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Theorem elelsuc 5766
 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 400 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
2 elsucg 5761 . 2 (𝐴𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2mpbird 247 1 (𝐴𝐵𝐴 ∈ suc 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   = wceq 1480   ∈ wcel 1987  suc csuc 5694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-sn 4156  df-suc 5698 This theorem is referenced by:  suctr  5777  suctrOLD  5778  pssnn  8138  pwsdompw  8986  fin1a2lem4  9185  grur1a  9601  bnj570  30736  finxpsuclem  32905
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