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Theorem eleldisjs 38189
Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
Assertion
Ref Expression
eleldisjs (𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ( E ↾ 𝐴) ∈ Disjs ))

Proof of Theorem eleldisjs
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reseq2 5974 . . 3 (𝑎 = 𝐴 → ( E ↾ 𝑎) = ( E ↾ 𝐴))
21eleq1d 2813 . 2 (𝑎 = 𝐴 → (( E ↾ 𝑎) ∈ Disjs ↔ ( E ↾ 𝐴) ∈ Disjs ))
3 df-eldisjs 38167 . 2 ElDisjs = {𝑎 ∣ ( E ↾ 𝑎) ∈ Disjs }
42, 3elab2g 3667 1 (𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ( E ↾ 𝐴) ∈ Disjs ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099   E cep 5575  ccnv 5671  cres 5674   Disjs cdisjs 37670   ElDisjs celdisjs 37672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-in 3951  df-opab 5205  df-xp 5678  df-res 5684  df-eldisjs 38167
This theorem is referenced by:  eleldisjseldisj  38190
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