Theorem nelb 3240

Description: A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelb

Step	Hyp	Ref	Expression
1		df-ne 2947	. . . 4 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
2	1	ralbii 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴)
3		ralnex 3078	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
4	2, 3	bitr2i 276	. 2 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
5		risset 3239	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
6	4, 5	xchnxbir 333	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077

This theorem is referenced by:  dfdif3  4140  inpr0  32560