eleldisjseldisj - Mathbox for Peter Mazsa

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Theorem eleldisjseldisj 38706

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 38705	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (^◡ E ↾ 𝐴) ∈ Disjs ))
2		cnvepresex 38303	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (^◡ E ↾ 𝐴) ∈ V)
3		eldisjsdisj 38704	. . . 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 38684	. 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 2109 Vcvv 3438 E cep 5522 ^◡ccnv 5622 ↾ cres 5625 Disjs cdisjs 38187 Disj wdisjALTV 38188 ElDisjs celdisjs 38189 ElDisj weldisj 38190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-eprel 5523 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-coss 38387 df-rels 38461 df-ssr 38474 df-cnvrefs 38501 df-cnvrefrels 38502 df-cnvrefrel 38503 df-disjss 38680 df-disjs 38681 df-disjALTV 38682 df-eldisjs 38683 df-eldisj 38684

This theorem is referenced by: (None)

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