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

Theorem sucel 5762
Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
sucel (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem sucel
StepHypRef Expression
1 risset 3056 . 2 (suc 𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = suc 𝐴)
2 dfcleq 2615 . . . 4 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴))
3 vex 3192 . . . . . . 7 𝑦 ∈ V
43elsuc 5758 . . . . . 6 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
54bibi2i 327 . . . . 5 ((𝑦𝑥𝑦 ∈ suc 𝐴) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
65albii 1744 . . . 4 (∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
72, 6bitri 264 . . 3 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
87rexbii 3035 . 2 (∃𝑥𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
91, 8bitri 264 1 (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wal 1478   = wceq 1480  wcel 1987  wrex 2908  suc csuc 5689
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-rex 2913  df-v 3191  df-un 3564  df-sn 4154  df-suc 5693
This theorem is referenced by:  axinf2  8489  zfinf2  8491
  Copyright terms: Public domain W3C validator