Theorem nelbOLD 3195

Description: Obsolete version of nelb 3194 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 2943	. . . . 5 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
2	1	ralbii 3090	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
3		ralnex 3163	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
4	2, 3	bitri 274	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
5		risset 3193	. . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
6	4, 5	xchbinxr 334	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)
7	6	bicomi 223	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069

This theorem is referenced by: (None)