eldisjsdisj - Mathbox for Peter Mazsa

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Theorem eldisjsdisj 39008

Description: The element of the class of disjoint relations and the disjoint relation predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)

**Assertion**
Ref	Expression
eldisjsdisj	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

**Proof of Theorem eldisjsdisj**
Step	Hyp	Ref	Expression
1		cosscnvex 38705	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ^◡𝑅 ∈ V)
2		elcnvrefrelsrel 38811	. . . 4 ⊢ ( ≀ ^◡𝑅 ∈ V → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
4		elrelsrel 38640	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
5	3, 4	anbi12d 632	. 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅)))
6		eldisjs 39003	. 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
7		df-disjALTV 38986	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
8	5, 6, 7	3bitr4g 314	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2113 Vcvv 3440 ^◡ccnv 5623 Rel wrel 5629 ≀ ccoss 38386 Rels crels 38388 CnvRefRels ccnvrefrels 38394 CnvRefRel wcnvrefrel 38395 Disjs cdisjs 38419 Disj wdisjALTV 38420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-rels 38638 df-coss 38696 df-ssr 38773 df-cnvrefs 38800 df-cnvrefrels 38801 df-cnvrefrel 38802 df-disjss 38984 df-disjs 38985 df-disjALTV 38986

This theorem is referenced by: eleldisjseldisj 39010 brpartspart 39054

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