| Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
| 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 39246 | . . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | |
| 2 | 1 | simplbi 499 | . . . 4 ⊢ ( Disj 𝑅 → ≀ ◡𝑅 ⊆ I ) |
| 3 | dfdisjALTV2 39246 | . . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ◡𝑆 ⊆ I ∧ Rel 𝑆)) | |
| 4 | 3 | simplbi 499 | . . . 4 ⊢ ( Disj 𝑆 → ≀ ◡𝑆 ⊆ I ) |
| 5 | 2, 4 | orim12i 917 | . . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I )) |
| 6 | inss 4195 | . . 3 ⊢ (( ≀ ◡𝑅 ⊆ I ∨ ≀ ◡𝑆 ⊆ I ) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
| 8 | disjxrn 39293 | . 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 856 ∩ cin 3898 ⊆ wss 3899 I cid 5534 ◡ccnv 5639 Rel wrel 5645 ⋉ cxrn 38621 ≀ ccoss 38630 Disj wdisjALTV 38666 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pr 5384 ax-un 7707 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-rab 3409 df-v 3450 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-fo 6516 df-fv 6518 df-1st 7959 df-2nd 7960 df-ec 8668 df-xrn 38827 df-coss 38948 df-cnvrefrel 39054 df-disjALTV 39237 |
| This theorem is referenced by: disjimxrn 39296 |
| Copyright terms: Public domain | W3C validator |