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Theorem elsuc 6010
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 6008 . 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 198  wo 874   = wceq 1653  wcel 2157  Vcvv 3385  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-un 3774  df-sn 4369  df-suc 5947
This theorem is referenced by:  sucel  6014  limsssuc  7284  omsmolem  7973  cantnfle  8818  infxpenlem  9122  inatsk  9888  untsucf  32102  dfon2lem7  32206  nolesgn2ores  32338
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