eldisjsdisj - Mathbox for Peter Mazsa

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Theorem eldisjsdisj 39163

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 38849	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ^◡𝑅 ∈ V)
2		elcnvrefrelsrel 38955	. . . 4 ⊢ ( ≀ ^◡𝑅 ∈ V → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
4		elrelsrel 38781	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
5	3, 4	anbi12d 633	. 2 ⊢ (𝑅 ∈ 𝑉 → (( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ) ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅)))
6		eldisjs 39158	. 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
7		df-disjALTV 39129	. 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 2114 Vcvv 3430 ^◡ccnv 5625 Rel wrel 5631 ≀ ccoss 38522 Rels crels 38524 CnvRefRels ccnvrefrels 38530 CnvRefRel wcnvrefrel 38531 Disjs cdisjs 38557 Disj wdisjALTV 38558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5232 ax-pow 5304 ax-pr 5372 ax-un 7684

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-rels 38779 df-coss 38840 df-ssr 38917 df-cnvrefs 38944 df-cnvrefrels 38945 df-cnvrefrel 38946 df-disjss 39127 df-disjs 39128 df-disjALTV 39129

This theorem is referenced by: qmapeldisjs 39164 eleldisjseldisj 39168 brpartspart 39215 eldisjsim1 39273 eldisjsim3 39276 eldisjs6 39279

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