Theorem sneqr 4152

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

Hypothesis

Ref	Expression
sneqr.1	|- A e. _V

Assertion

Ref	Expression
sneqr	|- ( { A } = { B } -> A = B )

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. . . 4 |- A e. _V
2	1	snid 4008	. . 3 |- A e. { A }
3		eleq2 2529	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 216	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsnc 4004	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 201	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   _Vcvv 3057   {csn 3980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sn 3981

This theorem is referenced by:  snsssn  4153  sneqrg  4154  opth1  4692  opthwiener  4720  canth2  7756  axcc2lem  8897  hashge3el3dif  12681  dis2ndc  20530  axlowdim1  25045  bj-snsetex  31603  poimirlem13  31999  poimirlem14  32000  wopprc  35931  hoidmv1le  38523  propeqop  39136  funsndifnop  39160