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Theorem elelsuc 4095
Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)
Assertion
Ref Expression
elelsuc (A BA suc B)

Proof of Theorem elelsuc
StepHypRef Expression
1 orc 620 . 2 (A B → (A B A = B))
2 elsucg 4090 . 2 (A B → (A suc B ↔ (A B A = B)))
31, 2mpbird 156 1 (A BA suc B)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228   wcel 1374  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:  suctrALT  4107  ordsuc  4225  nnaordex  6011
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