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Theorem elsuc2 4144
 Description: Membership in a successor. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1 𝐴 ∈ V
Assertion
Ref Expression
elsuc2 (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))

Proof of Theorem elsuc2
StepHypRef Expression
1 elsuc.1 . 2 𝐴 ∈ V
2 elsuc2g 4142 . 2 (𝐴 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
31, 2ax-mp 7 1 (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557  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:  nnsucelsuc  6070
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