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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelpr | Structured version Visualization version GIF version |
Description: A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
nelpr | ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 4652 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
2 | 1 | notbid 318 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
3 | neanior 3032 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | 2, 3 | bitr4di 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {cpr 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-un 3967 df-sn 4631 df-pr 4633 |
This theorem is referenced by: inpr0 32557 xnn01gt 32780 |
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