eldisjsdisj - Mathbox for Peter Mazsa

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Theorem eldisjsdisj 36450

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 36155	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ^◡𝑅 ∈ V)
2		elcnvrefrelsrel 36262	. . . 4 ⊢ ( ≀ ^◡𝑅 ∈ V → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
4		elrelsrel 36217	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
5	3, 4	anbi12d 634	. 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅)))
6		eldisjs 36445	. 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
7		df-disjALTV 36428	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2113 Vcvv 3397 ^◡ccnv 5518 Rel wrel 5524 ≀ ccoss 35945 Rels crels 35947 CnvRefRels ccnvrefrels 35953 CnvRefRel wcnvrefrel 35954 Disjs cdisjs 35978 Disj wdisjALTV 35979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-coss 36149 df-rels 36215 df-ssr 36228 df-cnvrefs 36253 df-cnvrefrels 36254 df-cnvrefrel 36255 df-disjss 36426 df-disjs 36427 df-disjALTV 36428

This theorem is referenced by: eleldisjseldisj 36452

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