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Theorem elelsuc 4146
 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 633 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 4141 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 156 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393  suc csuc 4102 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108 This theorem is referenced by:  suctr  4158  ordsuc  4287  nnaordex  6100
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