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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjs5 | Structured version Visualization version GIF version |
Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
eldisjs5 | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldisjs2 36873 | . 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels )) | |
2 | cosscnvssid5 36635 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)) | |
3 | elrelsrel 36644 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
4 | 3 | anbi2d 630 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅))) |
5 | 3 | anbi2d 630 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))) |
6 | 4, 5 | bibi12d 347 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )) ↔ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)))) |
7 | 2, 6 | mpbiri 259 | . 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ))) |
8 | 1, 7 | bitrid 284 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 845 = wceq 1539 ∈ wcel 2104 ∀wral 3062 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 I cid 5495 ◡ccnv 5595 dom cdm 5596 Rel wrel 5601 [cec 8523 ≀ ccoss 36374 Rels crels 36376 Disjs cdisjs 36407 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rmo 3285 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-ec 8527 df-coss 36576 df-rels 36642 df-ssr 36655 df-cnvrefs 36680 df-cnvrefrels 36681 df-disjss 36853 df-disjs 36854 |
This theorem is referenced by: (None) |
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