blob: 73d849f0fa0f49195fc895d50c48ef3a05b887b0 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
|
\begin{proposition}\label{bounded_fix_single}
Let $A$ be a set.
For all $x\in A$ we have $x\in A$.
\end{proposition}
\begin{proof}
Fix $x\in A$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{bounded_fix_multiple}
Let $A$ be a set.
For all $x,y\in A$ we have $x\in A$ and $y\in A$.
\end{proposition}
\begin{proof}
Fix $x,y\in A$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{bounded_fix_negative}
Let $A$ be a set.
For all $x\notin A$ we have $x\notin A$.
\end{proposition}
\begin{proof}
Fix $x\notin A$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{fix_such_that}
Let $A$ be a set.
For all $x$ such that $x\in A$ we have $x\in A$.
\end{proposition}
\begin{proof}
Fix $x$ such that $x\in A$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{assume_left_conjunct}
Let $A,B$ be sets.
If $A=A$ and $B=B$, then $A=A$.
\end{proposition}
\begin{proof}
Assume $A=A$.
Assume $B=B$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{assume_right_conjunct}
Let $A,B$ be sets.
If $A=A$ and $B=B$, then $B=B$.
\end{proposition}
\begin{proof}
Assume $B=B$.
Assume $A=A$.
Follows by assumption.
\end{proof}
\begin{proposition}\label{take_bounded}
Let $A$ be a set.
Suppose there exists $x\in A$ such that $x=x$.
Then $A=A$.
\end{proposition}
\begin{proof}
Take $x\in A$ such that $x=x$ by assumption.
We have $x\in A$ by assumption.
Follows.
\end{proof}
\begin{proposition}\label{take_named_noun}
Let $A$ be a set.
Suppose there exist sets $x,y$ such that $x=x$ and $y=y$.
Then $A=A$.
\end{proposition}
\begin{proof}
Take a set $x,y$ such that $x=x$ and $y=y$ by assumption.
Follows.
\end{proof}
\begin{proposition}\label{take_anonymous_noun}
Let $A$ be a set.
Suppose there exists a set.
Then $A=A$.
\end{proposition}
\begin{proof}
Take a set by assumption.
Follows.
\end{proof}
\begin{proposition}\label{existential_have_witness}
Let $A$ be a set.
Suppose there exists $x\in A$ such that $x=x$.
Then $A=A$.
\end{proposition}
\begin{proof}
We have there exists $x\in A$ such that $x=x$ by assumption.
We have $x\in A$ by assumption.
Follows.
\end{proof}
\begin{proposition}\label{take_omitted_continuation}
Let $A$ be a set.
Suppose there exists $x\in A$ such that $x=x$.
Then $A=A$.
\end{proposition}
\begin{proof}
Take $x\in A$ such that $x=x$ by assumption.
Omitted.
\end{proof}
|
