Theorem sucprcregOLD 9569

Description: Obsolete version of sucprcreg 9568 as of 31-Dec-2025. (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sucprcregOLD

Step	Hyp	Ref	Expression
1		sucprc 6440	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9562	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 6367	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2774	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 4150	. . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
6	4, 5	sylbb2 241	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7		snidg 4631	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8		ssel2 3940	. . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
9	6, 7, 8	syl2an 607	. . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴)
10	2, 9	mto 200	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
11	10	imnani 405	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
12	1, 11	impbii 212	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  {csn 4594  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-sn 4595  df-suc 6367

This theorem is referenced by: (None)