Theorem intsn 2568

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 2424	. . 3 |- {A} = {A, A}
2	1	inteqi 2541	. 2 |- |^|{A} = |^|{A, A}
3		intsn.1	. . 3 |- A e. V
4	3, 3	intpr 2567	. 2 |- |^|{A, A} = (A i^i A)
5		inidm 2225	. 2 |- (A i^i A) = A
6	2, 4, 5	3eqtr 1502	1 |- |^|{A} = A

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814   i^i cin 2049  {csn 2413  {cpr 2414  |^|cint 2537

This theorem is referenced by:  intunsn 2569  op1stb 2919  op2ndb 3457  cf0 4922  cflecard 4924  cfom 4928  fine 10442  abfi 10443  moec 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-in 2054  df-sn 2416  df-pr 2417  df-int 2538

Copyright terms: Public domain