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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 39295 | . . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | |
| 2 | 1 | simplbi 500 | . . . 4 ⊢ ( Disj 𝑅 → ≀ ◡𝑅 ⊆ I ) |
| 3 | dfdisjALTV2 39295 | . . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ◡𝑆 ⊆ I ∧ Rel 𝑆)) | |
| 4 | 3 | simplbi 500 | . . . 4 ⊢ ( Disj 𝑆 → ≀ ◡𝑆 ⊆ I ) |
| 5 | 2, 4 | orim12i 919 | . . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I )) |
| 6 | inss 4200 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I ) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
| 8 | disjxrn 39342 | . 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 858 ∩ cin 3903 ⊆ wss 3904 I cid 5541 ◡ccnv 5646 Rel wrel 5652 ⋉ cxrn 38670 ≀ ccoss 38679 Disj wdisjALTV 38715 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 ax-un 7718 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-fo 6527 df-fv 6529 df-1st 7970 df-2nd 7971 df-ec 8680 df-xrn 38876 df-coss 38997 df-cnvrefrel 39103 df-disjALTV 39286 |
| This theorem is referenced by: disjimxrn 39345 |
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