Theorem sneqr 4167

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 4026	. . 3 |- A e. { A }
3		eleq2 2496	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 214	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsnc 4022	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 199	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1437   e. wcel 1872   _Vcvv 3080   {csn 3998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sn 3999

This theorem is referenced by:  snsssn  4168  sneqrg  4169  opth1  4694  opthwiener  4722  canth2  7734  axcc2lem  8873  hashge3el3dif  12646  dis2ndc  20473  axlowdim1  24987  bj-snsetex  31525  poimirlem13  31917  poimirlem14  31918  wopprc  35855  hoidmv1le  38320  propeqop  38865  funsndifnop  38885