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Theorem sucel 4174
 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 2369 . 2 (suc 𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = suc 𝐴)
2 dfcleq 2050 . . . 4 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴))
3 vex 2577 . . . . . . 7 𝑦 ∈ V
43elsuc 4170 . . . . . 6 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
54bibi2i 220 . . . . 5 ((𝑦𝑥𝑦 ∈ suc 𝐴) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
65albii 1375 . . . 4 (∀𝑦(𝑦𝑥𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
72, 6bitri 177 . . 3 (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
87rexbii 2348 . 2 (∃𝑥𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
91, 8bitri 177 1 (suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   ∨ wo 639  ∀wal 1257   = wceq 1259   ∈ wcel 1409  ∃wrex 2324  suc csuc 4129 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-suc 4135 This theorem is referenced by: (None)
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