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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 2795 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ ¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | |
2 | opthg 4946 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
3 | 2 | notbid 308 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
4 | ianor 509 | . . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) | |
5 | df-ne 2795 | . . . . 5 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶) | |
6 | df-ne 2795 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷) | |
7 | 5, 6 | orbi12i 543 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷)) |
8 | 4, 7 | bitr4i 267 | . . 3 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
9 | 3, 8 | syl6bb 276 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) |
10 | 1, 9 | syl5bb 272 | 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 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 〈cop 4183 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 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-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: opthne 4951 zlmodzxznm 42286 |
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