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Theorem sucprcreg 9624
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)
Assertion
Ref Expression
sucprcreg 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Proof of Theorem sucprcreg
StepHypRef Expression
1 sucprc 6440 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 9620 . . . 4 ¬ 𝐴𝐴
3 df-suc 6370 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2730 . . . . . 6 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 4177 . . . . . 6 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5sylbb2 237 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7 snidg 4658 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8 ssel2 3967 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
96, 7, 8syl2an 594 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
102, 9mto 196 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1110imnani 399 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
121, 11impbii 208 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cun 3937  wss 3939  {csn 4624  suc csuc 6366
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-pr 5423  ax-reg 9615
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-sn 4625  df-pr 4627  df-suc 6370
This theorem is referenced by: (None)
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