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Theorem sucprc 5985
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 4410 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 207 . . 3 𝐴 ∈ V → {𝐴} = ∅)
32uneq2d 3931 . 2 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4 df-suc 5916 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 un0 4131 . . 3 (𝐴 ∪ ∅) = 𝐴
65eqcomi 2774 . 2 𝐴 = (𝐴 ∪ ∅)
73, 4, 63eqtr4g 2824 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  cun 3732  c0 4081  {csn 4336  suc csuc 5912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3737  df-un 3739  df-nul 4082  df-sn 4337  df-suc 5916
This theorem is referenced by:  nsuceq0  5990  sucon  7210  ordsuc  7216  sucprcreg  8717  suc11reg  8735
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