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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 3019 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ ¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
2 | opthg 5371 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
3 | 2 | notbid 320 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
4 | ianor 978 | . . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) | |
5 | df-ne 3019 | . . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶) | |
6 | df-ne 3019 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷) | |
7 | 5, 6 | orbi12i 911 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) |
8 | 4, 7 | bitr4i 280 | . . 3 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
9 | 3, 8 | syl6bb 289 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) |
10 | 1, 9 | syl5bb 285 | 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 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 〈cop 4575 |
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 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 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-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: opthne 5376 addsqnreup 26021 linds2eq 30943 zlmodzxznm 44559 |
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