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Theorem elsuc 5697
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 5695 . 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 194  wo 381   = wceq 1474  wcel 1976  Vcvv 3172  suc csuc 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-un 3544  df-sn 4125  df-suc 5632
This theorem is referenced by:  sucel  5701  suctrOLD  5712  limsssuc  6919  omsmolem  7597  cantnfle  8428  infxpenlem  8696  inatsk  9456  untsucf  30647  dfon2lem7  30744
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