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Theorem sucel 4302
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 2440 . 2 (suc 𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = suc 𝐴)
2 dfcleq 2111 . . . 4 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴))
3 vex 2663 . . . . . . 7 𝑦 ∈ V
43elsuc 4298 . . . . . 6 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
54bibi2i 226 . . . . 5 ((𝑦𝑥𝑦 ∈ suc 𝐴) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
65albii 1431 . . . 4 (∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
72, 6bitri 183 . . 3 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
87rexbii 2419 . 2 (∃𝑥𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
91, 8bitri 183 1 (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 682  wal 1314   = wceq 1316  wcel 1465  wrex 2394  suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-suc 4263
This theorem is referenced by: (None)
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