Theorem eleldisjs 38111

Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)

Assertion

Ref	Expression
eleldisjs	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Proof of Theorem eleldisjs

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reseq2 5970	. . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	eleq1d 2812	. 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
3		df-eldisjs 38089	. 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
4	2, 3	elab2g 3665	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   E cep 5572  ^◡ccnv 5668   ↾ cres 5671   Disjs cdisjs 37589   ElDisjs celdisjs 37591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-in 3950  df-opab 5204  df-xp 5675  df-res 5681  df-eldisjs 38089

This theorem is referenced by:  eleldisjseldisj  38112