Theorem sneqr 3761

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 3624	. . 3 |- A e. { A }
3		eleq2 2241	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 148	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsn 3609	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 122	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2738   {csn 3593

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-sn 3599

This theorem is referenced by:  sneqrg  3763  opth1  4237  cc2lem  7265