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Theorem sucprc 6445
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 4722 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 4162 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 6375 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4391 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2737 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2793 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  cun 3945  c0 4323  {csn 4629  suc csuc 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-dif 3950  df-un 3952  df-nul 4324  df-sn 4630  df-suc 6375
This theorem is referenced by:  nsuceq0  6452  sucon  7806  ordsuc  7816  ordsucOLD  7817  sucprcreg  9625  suc11reg  9643
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