Theorem sneqr 3747

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 3614	. . 3 |- A e. { A }
3		eleq2 2234	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 147	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsn 3599	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 121	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by:  sneqrg  3749  opth1  4221  cc2lem  7228