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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 5487 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 〈cop 4639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 |
This theorem is referenced by: xpord2lem 8156 xpord2pred 8159 xpord2indlem 8161 m2detleib 22624 addsqnreup 27472 mulsval 28110 zlmodzxzldeplem 47881 line2x 48142 inlinecirc02plem 48174 |
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