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Theorem sucprc 6460
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 4717 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 216 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 4168 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 6390 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4394 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2746 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2802 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-suc 6390
This theorem is referenced by:  nsuceq0  6467  sucon  7823  ordsuc  7833  ordsucOLD  7834  sucprcreg  9641  suc11reg  9659
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