Theorem sucprcreg 8506

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 5800	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 8505	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 5729	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2627	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 3786	. . . . . . 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 4206	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
9		ssel2 3598	. . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
10	7, 8, 9	syl2an 494	. . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴)
11	2, 10	mto 188	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
12	11	imnani 439	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
13	1, 12	impbii 199	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-suc 5729

This theorem is referenced by: (None)