Theorem sneqr 3866

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 3722	. . 3 |- A e. { A }
3		eleq2 2298	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 148	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsn 3707	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 122	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sn 3697

This theorem is referenced by:  sneqrg  3868  opth1  4354  cc2lem  7582