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

Theorem elelsuc 6438
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 866 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 6433 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 257 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wcel 2107  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-suc 6371
This theorem is referenced by:  suctr  6451  pssnn  9168  pssnnOLD  9265  ttrcltr  9711  ttrclss  9715  ttrclselem2  9721  pwsdompw  10199  fin1a2lem4  10398  grur1a  10814  bnj570  33916  satom  34347  satfv0  34349  satfvsuc  34352  satf00  34365  satf0suc  34367  sat1el2xp  34370  fmla  34372  fmla0  34373  fmlasuc0  34375  satfdmfmla  34391  finxpsuclem  36278
  Copyright terms: Public domain W3C validator