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Mirrors > Home > MPE Home > Th. List > opthneg | Structured version Visualization version GIF version |
Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.) |
Ref | Expression |
---|---|
opthneg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2782 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ ¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
2 | opthg 4872 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
3 | 2 | notbid 307 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
4 | ianor 508 | . . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) | |
5 | df-ne 2782 | . . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶) | |
6 | df-ne 2782 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷) | |
7 | 5, 6 | orbi12i 542 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) |
8 | 4, 7 | bitr4i 266 | . . 3 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
9 | 3, 8 | syl6bb 275 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) |
10 | 1, 9 | syl5bb 271 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: opthne 4877 zlmodzxznm 42080 |
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