eleldisjseldisj - Mathbox for Peter Mazsa

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Theorem eleldisjseldisj 36840

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 36839	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (^◡ E ↾ 𝐴) ∈ Disjs ))
2		cnvepresex 36469	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (^◡ E ↾ 𝐴) ∈ V)
3		eldisjsdisj 36838	. . . 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 278	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ Disj (^◡ E ↾ 𝐴)))
6		df-eldisj 36818	. 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 205 ∈ wcel 2106 Vcvv 3432 E cep 5494 ^◡ccnv 5588 ↾ cres 5591 Disjs cdisjs 36366 Disj wdisjALTV 36367 ElDisjs celdisjs 36368 ElDisj weldisj 36369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-eprel 5495 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-coss 36537 df-rels 36603 df-ssr 36616 df-cnvrefs 36641 df-cnvrefrels 36642 df-cnvrefrel 36643 df-disjss 36814 df-disjs 36815 df-disjALTV 36816 df-eldisjs 36817 df-eldisj 36818

This theorem is referenced by: (None)

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