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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdisjs5 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
| Ref | Expression |
|---|---|
| dfdisjs5 | ⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdisjs2 38710 | . 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } | |
| 2 | cosscnvssid5 38479 | . . 3 ⊢ (( ≀ ◡𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟)) | |
| 3 | elrelsrelim 38489 | . . . . 5 ⊢ (𝑟 ∈ Rels → Rel 𝑟) | |
| 4 | 3 | biantrud 531 | . . . 4 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ⊆ I ↔ ( ≀ ◡𝑟 ⊆ I ∧ Rel 𝑟))) |
| 5 | 3 | biantrud 531 | . . . 4 ⊢ (𝑟 ∈ Rels → (∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ↔ (∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟))) |
| 6 | 4, 5 | bibi12d 345 | . . 3 ⊢ (𝑟 ∈ Rels → (( ≀ ◡𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)) ↔ (( ≀ ◡𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟)))) |
| 7 | 2, 6 | mpbiri 258 | . 2 ⊢ (𝑟 ∈ Rels → ( ≀ ◡𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅))) |
| 8 | 1, 7 | rabimbieq 38252 | 1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 I cid 5577 ◡ccnv 5684 dom cdm 5685 Rel wrel 5690 [cec 8743 ≀ ccoss 38182 Rels crels 38184 Disjs cdisjs 38215 |
| 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-pow 5365 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-ral 3062 df-rex 3071 df-rmo 3380 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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 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-ec 8747 df-coss 38412 df-rels 38486 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 df-disjss 38704 df-disjs 38705 |
| This theorem is referenced by: (None) |
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