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Theorem elsuc 6253
 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 6251 . 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 208   ∨ wo 843   = wceq 1531   ∈ wcel 2108  Vcvv 3493  suc csuc 6186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-sn 4560  df-suc 6190 This theorem is referenced by:  sucel  6257  limsssuc  7557  omsmolem  8272  cantnfle  9126  infxpenlem  9431  inatsk  10192  untsucf  32929  dfon2lem7  33027  nolesgn2ores  33172  rdgssun  34651
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