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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 2800 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵) |
3 | elpri 4197 | . . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | 4 | ord 392 | . 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 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: fourierdlem70 40393 fourierdlem71 40394 fouriersw 40448 prsal 40538 sge0pr 40611 |
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