Theorem intsn 3963

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 3962	. 2 |- ( A e. _V -> |^| { A } = A )
3	1, 2	ax-mp 5	1 |- |^| { A } = A

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   |^|cint 3928

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-int 3929

This theorem is referenced by:  uniintsnr  3964  intunsn  3966  op1stb  4575  op2ndb  5220  ssfii  7173