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Theorem sucprc 4230
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 4189 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 snprc 3502 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 uneq2 3146 . . . 4 ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
42, 3sylbi 119 . . 3 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
51, 4syl5eq 2132 . 2 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6 un0 3314 . 2 (𝐴 ∪ ∅) = 𝐴
75, 6syl6eq 2136 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  cun 2995  c0 3284  {csn 3441  suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-suc 4189
This theorem is referenced by:  sucprcreg  4355  sucon  4359
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