eleldisjseldisj - Mathbox for Peter Mazsa

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Theorem eleldisjseldisj 38847

Description: The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set. (Contributed by Peter Mazsa, 23-Jul-2023.)

**Assertion**
Ref	Expression
eleldisjseldisj	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴))

**Proof of Theorem eleldisjseldisj**
Step	Hyp	Ref	Expression
1		eleldisjs 38846	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (^◡ E ↾ 𝐴) ∈ Disjs ))
2		cnvepresex 38388	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (^◡ E ↾ 𝐴) ∈ V)
3		eldisjsdisj 38845	. . . 4 ⊢ ((^◡ E ↾ 𝐴) ∈ V → ((^◡ E ↾ 𝐴) ∈ Disjs ↔ Disj (^◡ E ↾ 𝐴)))
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((^◡ E ↾ 𝐴) ∈ Disjs ↔ Disj (^◡ E ↾ 𝐴)))
5	1, 4	bitrd 279	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ Disj (^◡ E ↾ 𝐴)))
6		df-eldisj 38825	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (^◡ E ↾ 𝐴))
7	5, 6	bitr4di 289	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2113 Vcvv 3437 E cep 5518 ^◡ccnv 5618 ↾ cres 5621 Disjs cdisjs 38275 Disj wdisjALTV 38276 ElDisjs celdisjs 38277 ElDisj weldisj 38278

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-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

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-nf 1785 df-sb 2068 df-mo 2537 df-clab 2712 df-cleq 2725 df-clel 2808 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-eprel 5519 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-rels 38484 df-coss 38533 df-ssr 38610 df-cnvrefs 38637 df-cnvrefrels 38638 df-cnvrefrel 38639 df-disjss 38821 df-disjs 38822 df-disjALTV 38823 df-eldisjs 38824 df-eldisj 38825

This theorem is referenced by: (None)

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