Theorem nelbOLD 3233

Description: Obsolete version of nelb 3232 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 2939	. . . . 5 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
2	1	ralbii 3091	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
3		ralnex 3070	. . . 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 3231	. . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069

This theorem is referenced by: (None)