Theorem sneqr 4052

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 3917	. . 3 |- A e. { A }
3		eleq2 2504	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 211	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsnc 3913	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 196	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   _Vcvv 2984   {csn 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-sn 3890

This theorem is referenced by:  snsssn  4053  sneqrg  4054  opth1  4577  opthwiener  4605  canth2  7476  axcc2lem  8617  hashge3el3dif  12199  dis2ndc  19076  axlowdim1  23217  wopprc  29391  bj-snsetex  32468