Theorem sneqr 4199

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 4060	. . 3 |- A e. { A }
3		eleq2 2530	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 211	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsnc 4056	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 196	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 1819   _Vcvv 3109   {csn 4032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sn 4033

This theorem is referenced by:  snsssn  4200  sneqrg  4201  opth1  4729  opthwiener  4758  canth2  7689  axcc2lem  8833  hashge3el3dif  12527  dis2ndc  20086  axlowdim1  24388  wopprc  31134  bj-snsetex  34622