disjorimxrn - Mathbox for Peter Mazsa

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Theorem disjorimxrn 39780

Description: Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)

**Assertion**
Ref	Expression
disjorimxrn	⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆))

**Proof of Theorem disjorimxrn**
Step	Hyp	Ref	Expression
1		dfdisjALTV2 39731	. . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ^◡𝑅 ⊆ I ∧ Rel 𝑅))
2	1	simplbi 502	. . . 4 ⊢ ( Disj 𝑅 → ≀ ^◡𝑅 ⊆ I )
3		dfdisjALTV2 39731	. . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ^◡𝑆 ⊆ I ∧ Rel 𝑆))
4	3	simplbi 502	. . . 4 ⊢ ( Disj 𝑆 → ≀ ^◡𝑆 ⊆ I )
5	2, 4	orim12i 922	. . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ))
6		inss 4194	. . 3 ⊢ (( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
7	5, 6	syl 18	. 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
8		disjxrn 39778	. 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
9	7, 8	sylibr 237	1 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∨ wo 861 ∩ cin 3898 ⊆ wss 3899 I cid 5545 ^◡ccnv 5650 Rel wrel 5656 ⋉ cxrn 39106 ≀ ccoss 39115 Disj wdisjALTV 39151

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fo 6544 df-fv 6546 df-1st 8001 df-2nd 8002 df-ec 8719 df-xrn 39312 df-coss 39433 df-cnvrefrel 39539 df-disjALTV 39722

This theorem is used by: disjimxrn 39781

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