Theorem sucprcreg 9598

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 6434	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9594	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 6364	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2731	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 4178	. . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
6	4, 5	sylbb2 237	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7		snidg 4657	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8		ssel2 3972	. . . . 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 196	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
11	10	imnani 400	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
12	1, 11	impbii 208	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941   ⊆ wss 3943  {csn 4623  suc csuc 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-reg 9589

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-suc 6364

This theorem is referenced by: (None)