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Mirrors > Home > MPE Home > Th. List > Mathboxes > elprn2 | Structured version Visualization version GIF version |
Description: A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elprn2 | ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neneq 2951 | . . 3 ⊢ (𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 = 𝐶) |
3 | elpri 4589 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | orcom 867 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐵)) | |
6 | df-or 845 | . . . 4 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵)) | |
7 | 5, 6 | bitri 274 | . . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵)) |
8 | 4, 7 | sylib 217 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵)) |
9 | 2, 8 | mpd 15 | 1 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ne 2946 df-v 3433 df-un 3897 df-sn 4568 df-pr 4570 |
This theorem is referenced by: (None) |
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