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Theorem elsuc 4092
 Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1 A V
Assertion
Ref Expression
elsuc (A suc B ↔ (A B A = B))

Proof of Theorem elsuc
StepHypRef Expression
1 elsuc.1 . 2 A V
2 elsucg 4090 . 2 (A V → (A suc B ↔ (A B A = B)))
31, 2ax-mp 7 1 (A suc B ↔ (A B A = B))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374  Vcvv 2535  suc csuc 4051 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-un 2899  df-sn 3356  df-suc 4057 This theorem is referenced by:  sucel  4096  suctrALT  4107  tfrlemisucaccv  5860
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