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Theorem sucel 6146
 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 3233 . 2 (suc 𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = suc 𝐴)
2 dfcleq 2791 . . . 4 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴))
3 vex 3443 . . . . . . 7 𝑦 ∈ V
43elsuc 6142 . . . . . 6 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
54bibi2i 339 . . . . 5 ((𝑦𝑥𝑦 ∈ suc 𝐴) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
65albii 1805 . . . 4 (∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
72, 6bitri 276 . . 3 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
87rexbii 3213 . 2 (∃𝑥𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
91, 8bitri 276 1 (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∨ wo 842  ∀wal 1523   = wceq 1525   ∈ wcel 2083  ∃wrex 3108  suc csuc 6075 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-v 3442  df-un 3870  df-sn 4479  df-suc 6079 This theorem is referenced by:  axinf2  8956  zfinf2  8958
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