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Mirrors > Home > MPE Home > Th. List > nelb | Structured version Visualization version GIF version |
Description: A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) |
Ref | Expression |
---|---|
nelb | ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 3016 | . . . . 5 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴) | |
2 | 1 | ralbii 3164 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴) |
3 | ralnex 3235 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) |
5 | risset 3266 | . . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
6 | 4, 5 | xchbinxr 337 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵) |
7 | 6 | bicomi 226 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-clel 2892 df-ne 3016 df-ral 3142 df-rex 3143 |
This theorem is referenced by: inpr0 30290 |
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