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

Theorem elelsuc 6436
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 863 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 6431 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 256 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 843   = wceq 1539  wcel 2104  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-sn 4628  df-suc 6369
This theorem is referenced by:  suctr  6449  pssnn  9170  pssnnOLD  9267  ttrcltr  9713  ttrclss  9717  ttrclselem2  9723  pwsdompw  10201  fin1a2lem4  10400  grur1a  10816  bnj570  34214  satom  34645  satfv0  34647  satfvsuc  34650  satf00  34663  satf0suc  34665  sat1el2xp  34668  fmla  34670  fmla0  34671  fmlasuc0  34673  satfdmfmla  34689  finxpsuclem  36581
  Copyright terms: Public domain W3C validator