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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elprn1 | Structured version Visualization version GIF version | ||
| Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| elprn1 | ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neneq 2946 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵) |
| 3 | elpri 4649 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
| 5 | 4 | ord 865 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (¬ 𝐴 = 𝐵 → 𝐴 = 𝐶)) |
| 6 | 2, 5 | mpd 15 | 1 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → 𝐴 = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {cpr 4628 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: fourierdlem70 46191 fourierdlem71 46192 fouriersw 46246 prsal 46333 sge0pr 46409 |
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