disjorimxrn - Mathbox for Peter Mazsa

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Theorem disjorimxrn 39917

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 39868	. . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ^◡𝑅 ⊆ I ∧ Rel 𝑅))
2	1	simplbi 502	. . . 4 ⊢ ( Disj 𝑅 → ≀ ^◡𝑅 ⊆ I )
3		dfdisjALTV2 39868	. . . . 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 4144	. . 3 ⊢ (( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
7	5, 6	syl 18	. 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
8		disjxrn 39915	. 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 ⊆ wss 3896 ∩ cin 4088 I cid 5545 ^◡ccnv 5650 Rel wrel 5656 ⋉ cxrn 39243 ≀ ccoss 39252 Disj wdisjALTV 39288

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 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 6497 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 8005 df-2nd 8006 df-ec 8727 df-xrn 39449 df-coss 39570 df-cnvrefrel 39676 df-disjALTV 39859

This theorem is used by: disjimxrn 39918

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