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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 5336 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 846 ∈ wcel 2113 ≠ wne 2934 Vcvv 3397 〈cop 4519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3399 df-dif 3844 df-un 3846 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 |
This theorem is referenced by: m2detleib 21375 addsqnreup 26171 otthne 33248 xpord2lem 33392 xpord2pred 33395 xpord2ind 33397 zlmodzxzldeplem 45357 line2x 45618 inlinecirc02plem 45650 |
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