MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsuc2 Structured version   Visualization version   GIF version

Theorem elsuc2 5757
Description: Membership in a successor. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1 𝐴 ∈ V
Assertion
Ref Expression
elsuc2 (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))

Proof of Theorem elsuc2
StepHypRef Expression
1 elsuc.1 . 2 𝐴 ∈ V
2 elsuc2g 5755 . 2 (𝐴 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
31, 2ax-mp 5 1 (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383   = wceq 1480  wcel 1992  Vcvv 3191  suc csuc 5687
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565  df-sn 4154  df-suc 5691
This theorem is referenced by:  alephordi  8842
  Copyright terms: Public domain W3C validator