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Theorem sucprcreg 9641
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 6460 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 9637 . . . 4 ¬ 𝐴𝐴
3 df-suc 6390 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2742 . . . . . 6 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 4189 . . . . . 6 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5sylbb2 238 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7 snidg 4660 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8 ssel2 3978 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
96, 7, 8syl2an 596 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
102, 9mto 197 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1110imnani 400 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
121, 11impbii 209 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  {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-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-suc 6390
This theorem is referenced by: (None)
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