eleldisjseldisj - Mathbox for Peter Mazsa

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Theorem eleldisjseldisj 38705

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 38704	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (^◡ E ↾ 𝐴) ∈ Disjs ))
2		cnvepresex 38310	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (^◡ E ↾ 𝐴) ∈ V)
3		eldisjsdisj 38703	. . . 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 38683	. 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 2107 Vcvv 3463 E cep 5563 ^◡ccnv 5664 ↾ cres 5667 Disjs cdisjs 38190 Disj wdisjALTV 38191 ElDisjs celdisjs 38192 ElDisj weldisj 38193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-clab 2713 df-cleq 2726 df-clel 2808 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-eprel 5564 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-coss 38387 df-rels 38461 df-ssr 38474 df-cnvrefs 38501 df-cnvrefrels 38502 df-cnvrefrel 38503 df-disjss 38679 df-disjs 38680 df-disjALTV 38681 df-eldisjs 38682 df-eldisj 38683

This theorem is referenced by: (None)

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