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Theorem sucprc 6259
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 snprc 4645 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 218 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 4137 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 6190 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4342 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2828 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2879 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1530  wcel 2107  Vcvv 3493  cun 3932  c0 4289  {csn 4559  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-suc 6190
This theorem is referenced by:  nsuceq0  6264  sucon  7515  ordsuc  7521  sucprcreg  9057  suc11reg  9074
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