eldisjsdisj - Mathbox for Peter Mazsa

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Theorem eldisjsdisj 36765

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 36470	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ^◡𝑅 ∈ V)
2		elcnvrefrelsrel 36577	. . . 4 ⊢ ( ≀ ^◡𝑅 ∈ V → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
4		elrelsrel 36532	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
5	3, 4	anbi12d 630	. 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅)))
6		eldisjs 36760	. 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
7		df-disjALTV 36743	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
8	5, 6, 7	3bitr4g 313	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2108 Vcvv 3422 ^◡ccnv 5579 Rel wrel 5585 ≀ ccoss 36260 Rels crels 36262 CnvRefRels ccnvrefrels 36268 CnvRefRel wcnvrefrel 36269 Disjs cdisjs 36293 Disj wdisjALTV 36294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-coss 36464 df-rels 36530 df-ssr 36543 df-cnvrefs 36568 df-cnvrefrels 36569 df-cnvrefrel 36570 df-disjss 36741 df-disjs 36742 df-disjALTV 36743

This theorem is referenced by: eleldisjseldisj 36767

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