Theorem nelbOLD 3234

Description: Obsolete version of nelb 3233 as of 3-Nov-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof modification is discouraged.) (New usage 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-ne 2940	. . . . 5 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
2	1	ralbii 3092	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
3		ralnex 3071	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
4	2, 3	bitri 275	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
5		risset 3232	. . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
6	4, 5	xchbinxr 335	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)
7	6	bicomi 224	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070

This theorem is referenced by: (None)