disjorimxrn - Mathbox for Peter Mazsa

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Theorem disjorimxrn 38735

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 38701	. . . . 5 ⊢ ( Disj 𝑅 ↔ ( ≀ ^◡𝑅 ⊆ I ∧ Rel 𝑅))
2	1	simplbi 497	. . . 4 ⊢ ( Disj 𝑅 → ≀ ^◡𝑅 ⊆ I )
3		dfdisjALTV2 38701	. . . . 5 ⊢ ( Disj 𝑆 ↔ ( ≀ ^◡𝑆 ⊆ I ∧ Rel 𝑆))
4	3	simplbi 497	. . . 4 ⊢ ( Disj 𝑆 → ≀ ^◡𝑆 ⊆ I )
5	2, 4	orim12i 908	. . 3 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ))
6		inss 4213	. . 3 ⊢ (( ≀ ^◡𝑅 ⊆ I ∨ ≀ ^◡𝑆 ⊆ I ) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
7	5, 6	syl 17	. 2 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
8		disjxrn 38733	. 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ^◡𝑅 ∩ ≀ ^◡𝑆) ⊆ I )
9	7, 8	sylibr 234	1 ⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 847 ∩ cin 3915 ⊆ wss 3916 I cid 5534 ^◡ccnv 5639 Rel wrel 5645 ⋉ cxrn 38163 ≀ ccoss 38164 Disj wdisjALTV 38198

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 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 6515 df-fn 6516 df-f 6517 df-fo 6519 df-fv 6521 df-1st 7970 df-2nd 7971 df-ec 8675 df-xrn 38348 df-coss 38397 df-cnvrefrel 38513 df-disjALTV 38692

This theorem is referenced by: disjimxrn 38736

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