Theorem nelbOLDOLD 30693

Description: Obsolete version of nelb 3195 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelbOLDOLD

Step	Hyp	Ref	Expression
1		df-ral 3069	. 2 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
2		df-ne 2944	. . 3 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
3	2	ralbii 3091	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
4		dfclel 2819	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
5	4	notbii 323	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
6		alnex 1789	. . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
7		imnan 403	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
8		ancom 464	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
9	8	notbii 323	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
10	7, 9	bitr2i 279	. . . 4 ⊢ (¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
11	10	albii 1827	. . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
12	5, 6, 11	3bitr2i 302	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝑥 = 𝐴))
13	1, 3, 12	3bitr4ri 307	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112   ≠ wne 2943  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-clel 2818  df-ne 2944  df-ral 3069

This theorem is referenced by: (None)