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Theorem sucprcreg 8389
 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 5717 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 8388 . . . 4 ¬ 𝐴𝐴
3 df-suc 5646 . . . . . . . 8 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2615 . . . . . . 7 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 3748 . . . . . . 7 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5bitr4i 266 . . . . . 6 (suc 𝐴 = 𝐴 ↔ {𝐴} ⊆ 𝐴)
76biimpi 205 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
8 snidg 4153 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
9 ssel2 3563 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
107, 8, 9syl2an 493 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
112, 10mto 187 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1211imnani 438 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
131, 12impbii 198 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  {csn 4125  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-suc 5646 This theorem is referenced by: (None)
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