Theorem intsn 2561

Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.

Hypothesis

Ref	Expression
intsn.1	|- A e. V

Assertion

Ref	Expression
intsn	|- |^|{A} = A

Proof of Theorem intsn

Step	Hyp	Ref	Expression
1		dfsn2 2418	. . 3 |- {A} = {A, A}
2	1	inteqi 2534	. 2 |- |^|{A} = |^|{A, A}
3		intsn.1	. . 3 |- A e. V
4	3, 3	intpr 2560	. 2 |- |^|{A, A} = (A i^i A)
5		inidm 2220	. 2 |- (A i^i A) = A
6	2, 4, 5	3eqtr 1498	1 |- |^|{A} = A

Syntax hints:   = wceq 955   e. wcel 957  Vcvv 1809   i^i cin 2044  {csn 2407  {cpr 2408  |^|cint 2530

This theorem is referenced by:  intunsn 2562  op1stb 2910  op2ndb 3448  abfii3 4550  cf0 4897  cflecard 4899  cfom 4903  subbas2 7624  fine 10441  abfi 10442  moec 10450

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-v 1810  df-un 2048  df-in 2049  df-sn 2410  df-pr 2411  df-int 2531

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