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Theorem elsuc 5955
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 5953 . 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 196  wo 382   = wceq 1632  wcel 2139  Vcvv 3340  suc csuc 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-sn 4322  df-suc 5890
This theorem is referenced by:  sucel  5959  suctrOLD  5970  limsssuc  7215  omsmolem  7902  cantnfle  8741  infxpenlem  9026  inatsk  9792  untsucf  31894  dfon2lem7  31999  nolesgn2ores  32131
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