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Theorem elsuc 5711
 Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1 𝐴 ∈ V
Assertion
Ref Expression
elsuc (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem elsuc
StepHypRef Expression
1 elsuc.1 . 2 𝐴 ∈ V
2 elsucg 5709 . 2 (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2ax-mp 5 1 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-suc 5646 This theorem is referenced by:  sucel  5715  suctrOLD  5726  limsssuc  6942  omsmolem  7620  cantnfle  8451  infxpenlem  8719  inatsk  9479  untsucf  30841  dfon2lem7  30938
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