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Theorem sucprcreg 9052
 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 6235 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 9048 . . . 4 ¬ 𝐴𝐴
3 df-suc 6166 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2803 . . . . . 6 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 4110 . . . . . 6 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5sylbb2 241 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7 snidg 4559 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8 ssel2 3910 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
96, 7, 8syl2an 598 . . . 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 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  {csn 4525  suc csuc 6162 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-reg 9043 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-suc 6166 This theorem is referenced by: (None)
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