Theorem sucprcreg 9518

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.) (Proof shortened by SN, 22-Apr-2026.)

Assertion

Ref	Expression
sucprcreg	⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Proof of Theorem sucprcreg

Step	Hyp	Ref	Expression
1		sucprc 6395	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9512	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		snssg 4722	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐴 ↔ {𝐴} ⊆ 𝐴))
4	2, 3	mtbii 327	. . 3 ⊢ (𝐴 ∈ V → ¬ {𝐴} ⊆ 𝐴)
5		df-suc 6323	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	eqeq1i 2745	. . . 4 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
7		ssequn2 4125	. . . 4 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
8	6, 7	sylbb2 239	. . 3 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
9	4, 8	nsyl3 138	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
10	1, 9	impbii 210	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890  {csn 4562  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-sn 4563  df-suc 6323

This theorem is referenced by: (None)