Theorem intsn 4003

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 4002	. 2 |- ( A e. _V -> |^| { A } = A )
3	1, 2	ax-mp 5	1 |- |^| { A } = A

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3708   |^|cint 3968

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-sn 3714  df-pr 3715  df-int 3969

This theorem is referenced by:  uniintsnr  4004  intunsn  4006  op1stb  4622  op2ndb  5269  ssfii  7301