| 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 38390. (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 38390 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅))) | |
| 2 | orcom 871 | . . 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 38285 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) | |
| 5 | 4 | orbi1d 917 | . 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 848 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 E cep 5583 ◡ccnv 5684 [cec 8743 ⋉ cxrn 38181 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-ec 8747 df-xrn 38372 |
| This theorem is referenced by: disjsuc2 38392 |
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