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Theorem sucprcreg 8858
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 6101 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 8854 . . . 4 ¬ 𝐴𝐴
3 df-suc 6032 . . . . . . 7 suc 𝐴 = (𝐴 ∪ {𝐴})
43eqeq1i 2776 . . . . . 6 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5 ssequn2 4041 . . . . . 6 ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
64, 5sylbb2 230 . . . . 5 (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7 snidg 4467 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8 ssel2 3846 . . . . 5 (({𝐴} ⊆ 𝐴𝐴 ∈ {𝐴}) → 𝐴𝐴)
96, 7, 8syl2an 587 . . . 4 ((suc 𝐴 = 𝐴𝐴 ∈ V) → 𝐴𝐴)
102, 9mto 189 . . 3 ¬ (suc 𝐴 = 𝐴𝐴 ∈ V)
1110imnani 392 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
121, 11impbii 201 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 387   = wceq 1508  wcel 2051  Vcvv 3408  cun 3820  wss 3822  {csn 4435  suc csuc 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-reg 8849
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-sn 4436  df-pr 4438  df-suc 6032
This theorem is referenced by: (None)
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