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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 3024 | . . 3 ⊢ (𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 = 𝐶) |
3 | elpri 4591 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | orcom 866 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐵)) | |
6 | df-or 844 | . . . 4 ⊢ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐵) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵)) | |
7 | 5, 6 | bitri 277 | . . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ↔ (¬ 𝐴 = 𝐶 → 𝐴 = 𝐵)) |
8 | 4, 7 | sylib 220 | . 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 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-un 3943 df-sn 4570 df-pr 4572 |
This theorem is referenced by: (None) |
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