Theorem sucprcreg 9578

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 6446	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9572	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		snssg 4754	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐴 ↔ {𝐴} ⊆ 𝐴))
4	2, 3	mtbii 329	. . 3 ⊢ (𝐴 ∈ V → ¬ {𝐴} ⊆ 𝐴)
5		df-suc 6373	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	eqeq1i 2771	. . . 4 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
7		ssequn2 4145	. . . 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)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∪ cun 3906   ⊆ wss 3908  {csn 4594  suc csuc 6369

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-sn 4595  df-suc 6373

This theorem is used by: (None)