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Theorem elsuc2 6234
 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 6232 . 2 (𝐴 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
31, 2ax-mp 5 1 (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2115  Vcvv 3471  suc csuc 6166 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-un 3915  df-sn 4541  df-suc 6170 This theorem is referenced by:  alephordi  9477
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