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Theorem sucel 6438
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 3229 . 2 (suc 𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = suc 𝐴)
2 dfcleq 2724 . . . 4 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴))
3 vex 3477 . . . . . . 7 𝑦 ∈ V
43elsuc 6434 . . . . . 6 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
54bibi2i 337 . . . . 5 ((𝑦𝑥𝑦 ∈ suc 𝐴) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
65albii 1820 . . . 4 (∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
72, 6bitri 275 . . 3 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
87rexbii 3093 . 2 (∃𝑥𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
91, 8bitri 275 1 (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844  wal 1538   = wceq 1540  wcel 2105  wrex 3069  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-un 3953  df-sn 4629  df-suc 6370
This theorem is referenced by:  axinf2  9641  zfinf2  9643
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