Theorem sucprcreg 8544

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.)

Assertion

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

Proof of Theorem sucprcreg

Step	Hyp	Ref	Expression
1		sucprc 5838	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 8543	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 5767	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2656	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 3819	. . . . . . 7 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
6	4, 5	bitr4i 267	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ {𝐴} ⊆ 𝐴)
7	6	biimpi 206	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
8		snidg 4239	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
9		ssel2 3631	. . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
10	7, 8, 9	syl2an 493	. . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴)
11	2, 10	mto 188	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
12	11	imnani 438	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
13	1, 12	impbii 199	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210  suc csuc 5763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-suc 5767

This theorem is referenced by: (None)