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Theorem sucprcreg 9217
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 6288 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 9213 . . . 4 ¬ 𝐴𝐴
3 df-suc 6219 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2742 . . . . . 6 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 4097 . . . . . 6 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5sylbb2 241 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7 snidg 4575 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8 ssel2 3895 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
96, 7, 8syl2an 599 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
102, 9mto 200 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1110imnani 404 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
121, 11impbii 212 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  cun 3864  wss 3866  {csn 4541  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-reg 9208
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-sn 4542  df-pr 4544  df-suc 6219
This theorem is referenced by: (None)
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