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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 37143 | . . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | |
2 | 1 | simplbi 498 | . . . 4 ⊢ ( Disj 𝑅 → ≀ ◡𝑅 ⊆ I ) |
3 | dfdisjALTV2 37143 | . . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ◡𝑆 ⊆ I ∧ Rel 𝑆)) | |
4 | 3 | simplbi 498 | . . . 4 ⊢ ( Disj 𝑆 → ≀ ◡𝑆 ⊆ I ) |
5 | 2, 4 | orim12i 907 | . . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I )) |
6 | inss 4196 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I ) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
8 | disjxrn 37175 | . 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 845 ∩ cin 3907 ⊆ wss 3908 I cid 5528 ◡ccnv 5630 Rel wrel 5636 ⋉ cxrn 36600 ≀ ccoss 36601 Disj wdisjALTV 36635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fo 6499 df-fv 6501 df-1st 7917 df-2nd 7918 df-ec 8646 df-xrn 36800 df-coss 36840 df-cnvrefrel 36956 df-disjALTV 37134 |
This theorem is referenced by: disjimxrn 37178 |
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