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Theorem sucprc 4175
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 4134 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 snprc 3465 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 uneq2 3121 . . . 4 ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
42, 3sylbi 119 . . 3 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
51, 4syl5eq 2126 . 2 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6 un0 3285 . 2 (𝐴 ∪ ∅) = 𝐴
75, 6syl6eq 2130 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wcel 1434  Vcvv 2602  cun 2972  c0 3258  {csn 3406  suc csuc 4128
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-suc 4134
This theorem is referenced by:  sucprcreg  4300  sucon  4304
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