Theorem sucprcreg 9670

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 6471	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9666	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 6401	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2745	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 4212	. . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
6	4, 5	sylbb2 238	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7		snidg 4682	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8		ssel2 4003	. . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
9	6, 7, 8	syl2an 595	. . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴)
10	2, 9	mto 197	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
11	10	imnani 400	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
12	1, 11	impbii 209	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  {csn 4648  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-suc 6401

This theorem is referenced by: (None)