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Mirrors > Home > MPE Home > Th. List > Mathboxes > otthne | Structured version Visualization version GIF version |
Description: Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
otthne.1 | ⊢ 𝐴 ∈ V |
otthne.2 | ⊢ 𝐵 ∈ V |
otthne.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
otthne | ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ≠ 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | otthne.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | otthne.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opthne 5382 | . . 3 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐷, 𝐸〉 ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸)) |
4 | 3 | orbi1i 914 | . 2 ⊢ ((〈𝐴, 𝐵〉 ≠ 〈𝐷, 𝐸〉 ∨ 𝐶 ≠ 𝐹) ↔ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸) ∨ 𝐶 ≠ 𝐹)) |
5 | opex 5364 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
6 | otthne.3 | . . 3 ⊢ 𝐶 ∈ V | |
7 | 5, 6 | opthne 5382 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ≠ 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 ≠ 〈𝐷, 𝐸〉 ∨ 𝐶 ≠ 𝐹)) |
8 | df-3or 1090 | . 2 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹) ↔ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸) ∨ 𝐶 ≠ 𝐹)) | |
9 | 4, 7, 8 | 3bitr4i 306 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ≠ 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 847 ∨ w3o 1088 ∈ wcel 2112 ≠ wne 2943 Vcvv 3423 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 |
This theorem is referenced by: xpord3lem 33565 xpord3pred 33568 xpord3ind 33570 |
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