Theorem sucprcreg 9604

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 6441	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9598	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		snssg 4740	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐴 ↔ {𝐴} ⊆ 𝐴))
4	2, 3	mtbii 329	. . 3 ⊢ (𝐴 ∈ V → ¬ {𝐴} ⊆ 𝐴)
5		df-suc 6368	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	5	eqeq1i 2765	. . . 4 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
7		ssequn2 4187	. . . 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 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  {csn 4584  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-reg 9590

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-suc 6368

This theorem is used by: (None)