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Theorem sucprc 4098
 Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
Assertion
Ref Expression
sucprc A V → suc A = A)

Proof of Theorem sucprc
StepHypRef Expression
1 df-suc 4057 . . 3 suc A = (A ∪ {A})
2 snprc 3409 . . . 4 A V ↔ {A} = ∅)
3 uneq2 3068 . . . 4 ({A} = ∅ → (A ∪ {A}) = (A ∪ ∅))
42, 3sylbi 114 . . 3 A V → (A ∪ {A}) = (A ∪ ∅))
51, 4syl5eq 2066 . 2 A V → suc A = (A ∪ ∅))
6 un0 3228 . 2 (A ∪ ∅) = A
75, 6syl6eq 2070 1 A V → suc A = A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∪ cun 2892  ∅c0 3201  {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-in1 532  ax-in2 533  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-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-suc 4057 This theorem is referenced by:  sucprcreg  4211  sucon  4215
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