Theorem sucprcreg 9556

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 6428	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9550	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		snssg 4745	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐴 ↔ {𝐴} ⊆ 𝐴))
4	2, 3	mtbii 329	. . 3 ⊢ (𝐴 ∈ V → ¬ {𝐴} ⊆ 𝐴)
5		df-suc 6356	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	eqeq1i 2770	. . . 4 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
7		ssequn2 4144	. . . 4 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
8	6, 7	sylbb2 241	. . 3 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
9	4, 8	nsyl3 139	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
10	1, 9	impbii 212	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  {csn 4585  suc csuc 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-reg 9542

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-sn 4586  df-suc 6356

This theorem is referenced by: (None)