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

Theorem elsuc2 6037
 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 6035 . 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 198   ∨ wo 878   = wceq 1656   ∈ wcel 2164  Vcvv 3414  suc csuc 5969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803  df-sn 4400  df-suc 5973 This theorem is referenced by:  alephordi  9217
 Copyright terms: Public domain W3C validator