MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elelsuc Structured version   Visualization version   GIF version

Theorem elelsuc 5714
Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)
Assertion
Ref Expression
elelsuc (𝐴𝐵𝐴 ∈ suc 𝐵)

Proof of Theorem elelsuc
StepHypRef Expression
1 orc 399 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 5709 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 246 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1475  wcel 1977  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:  suctr  5725  suctrOLD  5726  pssnn  8063  pwsdompw  8909  fin1a2lem4  9108  grur1a  9520  bnj570  30229  finxpsuclem  32410
  Copyright terms: Public domain W3C validator