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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 5365 | . 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 208 ∨ wo 843 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 〈cop 4566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 |
This theorem is referenced by: m2detleib 21234 addsqnreup 26013 zlmodzxzldeplem 44547 line2x 44735 inlinecirc02plem 44767 |
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