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Theorem sucprc 4149
 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 4108 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 snprc 3435 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 uneq2 3091 . . . 4 ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
42, 3sylbi 114 . . 3 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
51, 4syl5eq 2084 . 2 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6 un0 3251 . 2 (𝐴 ∪ ∅) = 𝐴
75, 6syl6eq 2088 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  ∅c0 3224  {csn 3375  suc csuc 4102 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-suc 4108 This theorem is referenced by:  sucprcreg  4273  sucon  4277
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