Theorem nelbOLD 30232

Description: Obsolete version of nelb 3268 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
nelbOLD	⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelbOLD

Step	Hyp	Ref	Expression
1		df-ral 3143	. 2 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
2		df-ne 3017	. . 3 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
3	2	ralbii 3165	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
4		dfclel 2894	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
5	4	notbii 322	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
6		alnex 1782	. . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
7		imnan 402	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
8		ancom 463	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	notbii 322	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
10	7, 9	bitr2i 278	. . . 4 ⊢ (¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
11	10	albii 1820	. . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
12	5, 6, 11	3bitr2i 301	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
13	1, 3, 12	3bitr4ri 306	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-clel 2893  df-ne 3017  df-ral 3143

This theorem is referenced by: (None)