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Theorem elelsuc 4342
 Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)
Assertion
Ref Expression
elelsuc (𝐴𝐵𝐴 ∈ suc 𝐵)

Proof of Theorem elelsuc
StepHypRef Expression
1 orc 702 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 4337 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 166 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 698   = wceq 1332   ∈ wcel 2112  suc csuc 4298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-un 3082  df-sn 3540  df-suc 4304 This theorem is referenced by:  suctr  4354  ordsuc  4489  nnaordex  6435  fiintim  6834  exmidfodomrlemr  7087  exmidfodomrlemrALT  7088  3nelsucpw1  7114  ennnfonelemex  11999
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