Theorem sneqr 3790

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 3653	. . 3 |- A e. { A }
3		eleq2 2260	. . 3 |- ( { A } = { B } -> ( A e. { A } <-> A e. { B } ) )
4	2, 3	mpbii 148	. 2 |- ( { A } = { B } -> A e. { B } )
5	1	elsn 3638	. 2 |- ( A e. { B } <-> A = B )
6	4, 5	sylib 122	1 |- ( { A } = { B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by:  sneqrg  3792  opth1  4269  cc2lem  7333