Theorem intsn 3801

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 3800	. 2 |- ( A e. _V -> |^| { A } = A )
3	1, 2	ax-mp 5	1 |- |^| { A } = A

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2681   {csn 3522   |^|cint 3766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-sn 3528  df-pr 3529  df-int 3767

This theorem is referenced by:  uniintsnr  3802  intunsn  3804  op1stb  4394  op2ndb  5017  ssfii  6855