Theorem sucprcreg 9630

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 6462	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9624	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		snssg 4744	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐴 ↔ {𝐴} ⊆ 𝐴))
4	2, 3	mtbii 329	. . 3 ⊢ (𝐴 ∈ V → ¬ {𝐴} ⊆ 𝐴)
5		df-suc 6389	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	eqeq1i 2768	. . . 4 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
7		ssequn2 4190	. . . 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 2147  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588  suc csuc 6385

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-reg 9616

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-suc 6389

This theorem is used by: (None)