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Theorem sucel 4112
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 2346 . 2 (suc A Bx B x = suc A)
2 dfcleq 2031 . . . 4 (x = suc Ay(y xy suc A))
3 vex 2554 . . . . . . 7 y V
43elsuc 4108 . . . . . 6 (y suc A ↔ (y A y = A))
54bibi2i 216 . . . . 5 ((y xy suc A) ↔ (y x ↔ (y A y = A)))
65albii 1356 . . . 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 2325 . 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 628  wal 1240   = wceq 1242   wcel 1390  wrex 2301  suc csuc 4067
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3372  df-suc 4073
This theorem is referenced by: (None)
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