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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjecxrncnvep | Structured version Visualization version GIF version |
Description: Two ways of saying that cosets are disjoint, special case of disjecxrn 38371. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 25-Aug-2023.) |
Ref | Expression |
---|---|
disjecxrncnvep | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝐴 ∩ 𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjecxrn 38371 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅))) | |
2 | orcom 870 | . . 3 ⊢ ((([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅) ↔ (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)) | |
3 | 1, 2 | bitrdi 287 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))) |
4 | disjeccnvep 38266 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) | |
5 | 4 | orbi1d 916 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅) ↔ ((𝐴 ∩ 𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))) |
6 | 3, 5 | bitrd 279 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝐴 ∩ 𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 E cep 5588 ◡ccnv 5688 [cec 8742 ⋉ cxrn 38161 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-ec 8746 df-xrn 38353 |
This theorem is referenced by: disjsuc2 38373 |
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