eldisjsdisj - Mathbox for Peter Mazsa

	Mathbox for Peter Mazsa		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjsdisj		Structured version Visualization version GIF version

Theorem eldisjsdisj 35975

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 35680	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ≀ ^◡𝑅 ∈ V)
2		elcnvrefrelsrel 35787	. . . 4 ⊢ ( ≀ ^◡𝑅 ∈ V → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ≀ ^◡𝑅 ∈ CnvRefRels ↔ CnvRefRel ≀ ^◡𝑅))
4		elrelsrel 35742	. . 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 35970	. 2 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ^◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
7		df-disjALTV 35953	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
8	5, 6, 7	3bitr4g 316	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ^◡ccnv 5554 Rel wrel 5560 ≀ ccoss 35468 Rels crels 35470 CnvRefRels ccnvrefrels 35476 CnvRefRel wcnvrefrel 35477 Disjs cdisjs 35501 Disj wdisjALTV 35502

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-coss 35674 df-rels 35740 df-ssr 35753 df-cnvrefs 35778 df-cnvrefrels 35779 df-cnvrefrel 35780 df-disjss 35951 df-disjs 35952 df-disjALTV 35953

This theorem is referenced by: eleldisjseldisj 35977

W3C validator