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Theorem sneqr 3505
 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 V
Assertion
Ref Expression
sneqr ({A} = {B} → A = B)

Proof of Theorem sneqr
StepHypRef Expression
1 sneqr.1 . . . 4 A V
21snid 3377 . . 3 A {A}
3 eleq2 2083 . . 3 ({A} = {B} → (A {A} ↔ A {B}))
42, 3mpbii 136 . 2 ({A} = {B} → A {B})
51elsnc 3373 . 2 (A {B} ↔ A = B)
64, 5sylib 127 1 ({A} = {B} → A = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {csn 3350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sn 3356 This theorem is referenced by:  sneqrg  3507  opth1  3947
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