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Mirrors > Home > MPE Home > Th. List > opthne | 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 |
---|---|
opthne.1 | ⊢ 𝐴 ∈ V |
opthne.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opthne | ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthne.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opthne.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opthneg 5230 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) | |
4 | 1, 2, 3 | mp2an 679 | 1 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∨ wo 833 ∈ wcel 2050 ≠ wne 2968 Vcvv 3416 〈cop 4447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 |
This theorem is referenced by: m2detleib 20944 addsqnreup 25721 zlmodzxzldeplem 43918 line2x 44107 inlinecirc02plem 44139 |
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