Theorem sneqr 4197

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

Assertion

Ref	Expression
sneqr

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. . . 4
2	1	snid 4057	. . 3
3		eleq2 2530	. . 3
4	2, 3	mpbii 211	. 2
5	1	elsnc 4053	. 2
6	4, 5	sylib 196	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  cvv 3109  {csn 4029

This theorem is referenced by:  snsssn  4198  sneqrg  4199  opth1  4725  opthwiener  4754  canth2  7690  axcc2lem  8837  hashge3el3dif  12524  dis2ndc  19961  axlowdim1  24262  wopprc  30972  bj-snsetex  34521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sn 4030