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Theorem sucexb 4173
Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
Assertion
Ref Expression
sucexb (A V ↔ suc A V)

Proof of Theorem sucexb
StepHypRef Expression
1 unexb 4127 . 2 ((A V {A} V) ↔ (A ∪ {A}) V)
2 snexgOLD 3909 . . 3 (A V → {A} V)
32pm4.71i 371 . 2 (A V ↔ (A V {A} V))
4 df-suc 4057 . . 3 suc A = (A ∪ {A})
54eleq1i 2085 . 2 (suc A V ↔ (A ∪ {A}) V)
61, 3, 53bitr4i 201 1 (A V ↔ suc A V)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1374  Vcvv 2535  cun 2892  {csn 3350  suc csuc 4051
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-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
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 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-suc 4057
This theorem is referenced by:  sucexg  4174  sucelon  4179  onsucelsucr  4183  sucunielr  4185  peano2b  4264
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