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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 2328 . 2 (suc A Bx B x = suc A)
2 dfcleq 2016 . . . 4 (x = suc Ay(y xy suc A))
3 vex 2536 . . . . . . 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 2307 . 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 1374  ∃wrex 2283  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-un 2897  df-sn 3354  df-suc 4055 This theorem is referenced by: (None)
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