Theorem intsn 3880

Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Hypothesis

Ref	Expression
intsn.1	|- A e. _V

Assertion

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

Proof of Theorem intsn

Step	Hyp	Ref	Expression
1		intsn.1	. 2 |- A e. _V
2		intsng 3879	. 2 |- ( A e. _V -> |^| { A } = A )
3	1, 2	ax-mp 5	1 |- |^| { A } = A

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2738   {csn 3593   |^|cint 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-un 3134  df-in 3136  df-sn 3599  df-pr 3600  df-int 3846

This theorem is referenced by:  uniintsnr  3881  intunsn  3883  op1stb  4479  op2ndb  5113  ssfii  6973