Theorem sucprcregOLD 9555

Description: Obsolete version of sucprcreg 9554 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 6424	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 9548	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		df-suc 6352	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	eqeq1i 2767	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
5		ssequn2 4141	. . . . . 6 ⊢ ({𝐴} ⊆ 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
6	4, 5	sylbb2 240	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → {𝐴} ⊆ 𝐴)
7		snidg 4619	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
8		ssel2 3931	. . . . 5 ⊢ (({𝐴} ⊆ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
9	6, 7, 8	syl2an 605	. . . 4 ⊢ ((suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝐴)
10	2, 9	mto 199	. . 3 ⊢ ¬ (suc 𝐴 = 𝐴 ∧ 𝐴 ∈ V)
11	10	imnani 404	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
12	1, 11	impbii 211	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  {csn 4582  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-reg 9540

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4583  df-suc 6352

This theorem is referenced by: (None)