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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 35980 | . . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | |
2 | 1 | simplbi 500 | . . . 4 ⊢ ( Disj 𝑅 → ≀ ◡𝑅 ⊆ I ) |
3 | dfdisjALTV2 35980 | . . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ◡𝑆 ⊆ I ∧ Rel 𝑆)) | |
4 | 3 | simplbi 500 | . . . 4 ⊢ ( Disj 𝑆 → ≀ ◡𝑆 ⊆ I ) |
5 | 2, 4 | orim12i 905 | . . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I )) |
6 | inss 4208 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I ) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
8 | disjxrn 36010 | . 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
9 | 7, 8 | sylibr 236 | 1 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∩ cin 3928 ⊆ wss 3929 I cid 5452 ◡ccnv 5547 Rel wrel 5553 ⋉ cxrn 35485 ≀ ccoss 35486 Disj wdisjALTV 35520 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7682 df-2nd 7683 df-ec 8284 df-xrn 35656 df-coss 35692 df-cnvrefrel 35798 df-disjALTV 35971 |
This theorem is referenced by: disjimxrn 36012 |
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