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Theorem sucel 4094
Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
sucel (suc A Bx B y(y x ↔ (y A y = A)))
Distinct variable groups:   x,y,A   x,B
Allowed substitution hint:   B(y)

Proof of Theorem sucel
StepHypRef Expression
1 risset 2329 . 2 (suc A Bx B x = suc A)
2 dfcleq 2017 . . . 4 (x = suc Ay(y xy suc A))
3 vex 2537 . . . . . . 7 y V
43elsuc 4090 . . . . . 6 (y suc A ↔ (y A y = A))
54bibi2i 216 . . . . 5 ((y xy suc A) ↔ (y x ↔ (y A y = A)))
65albii 1339 . . . 4 (y(y xy suc A) ↔ y(y x ↔ (y A y = A)))
72, 6bitri 173 . . 3 (x = suc Ay(y x ↔ (y A y = A)))
87rexbii 2308 . 2 (x B x = suc Ax B y(y x ↔ (y A y = A)))
91, 8bitri 173 1 (suc A Bx B y(y x ↔ (y A y = A)))
Colors of variables: wff set class
Syntax hints:  wb 98   wo 616  wal 1226   = wceq 1228   wcel 1375  wrex 2284  suc csuc 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-rex 2289  df-v 2536  df-un 2898  df-sn 3355  df-suc 4055
This theorem is referenced by: (None)
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