| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eleldisjs | Structured version Visualization version GIF version | ||
| Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.) |
| Ref | Expression |
|---|---|
| eleldisjs | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reseq2 5990 | . . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴)) | |
| 2 | 1 | eleq1d 2825 | . 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) |
| 3 | df-eldisjs 38685 | . 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs } | |
| 4 | 2, 3 | elab2g 3679 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 E cep 5581 ◡ccnv 5682 ↾ cres 5685 Disjs cdisjs 38193 ElDisjs celdisjs 38195 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-in 3957 df-opab 5204 df-xp 5689 df-res 5695 df-eldisjs 38685 |
| This theorem is referenced by: eleldisjseldisj 38708 |
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