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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjorimxrn | Structured version Visualization version GIF version |
Description: Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
Ref | Expression |
---|---|
disjorimxrn | ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjALTV2 37226 | . . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | |
2 | 1 | simplbi 499 | . . . 4 ⊢ ( Disj 𝑅 → ≀ ◡𝑅 ⊆ I ) |
3 | dfdisjALTV2 37226 | . . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ◡𝑆 ⊆ I ∧ Rel 𝑆)) | |
4 | 3 | simplbi 499 | . . . 4 ⊢ ( Disj 𝑆 → ≀ ◡𝑆 ⊆ I ) |
5 | 2, 4 | orim12i 908 | . . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I )) |
6 | inss 4202 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I ) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
8 | disjxrn 37258 | . 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
9 | 7, 8 | sylibr 233 | 1 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 ∩ cin 3913 ⊆ wss 3914 I cid 5534 ◡ccnv 5636 Rel wrel 5642 ⋉ cxrn 36683 ≀ ccoss 36684 Disj wdisjALTV 36718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-1st 7925 df-2nd 7926 df-ec 8656 df-xrn 36883 df-coss 36923 df-cnvrefrel 37039 df-disjALTV 37217 |
This theorem is referenced by: disjimxrn 37261 |
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