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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelpr1 | Structured version Visualization version GIF version |
Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
nelpr1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
nelpr1.n | ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Ref | Expression |
---|---|
nelpr1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 399 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
3 | nelpr1.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | elprg 4144 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
7 | 2, 6 | mpbird 246 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶}) |
8 | nelpr1.n | . . . 4 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
10 | 7, 9 | pm2.65da 598 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
11 | 10 | neqned 2789 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: ovnsubadd2lem 39535 |
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