disjorimxrn - Mathbox for Peter Mazsa

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Theorem disjorimxrn 39555

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 39506	. . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ^◡𝑅 ⊆ I ∧ Rel 𝑅))
2	1	simplbi 502	. . . 4 ⊢ ( Disj 𝑅 → ≀ ^◡𝑅 ⊆ I )
3		dfdisjALTV2 39506	. . . . 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 4201	. . 3 ⊢ (( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
7	5, 6	syl 18	. 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
8		disjxrn 39553	. 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 3905 ⊆ wss 3906 I cid 5557 ^◡ccnv 5662 Rel wrel 5668 ⋉ cxrn 38881 ≀ ccoss 38890 Disj wdisjALTV 38926

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-un 7742

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-1st 7992 df-2nd 7993 df-ec 8702 df-xrn 39087 df-coss 39208 df-cnvrefrel 39314 df-disjALTV 39497

This theorem is used by: disjimxrn 39556

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