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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elscottab | Structured version Visualization version GIF version | ||
| Description: An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| elscottab.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| elscottab | ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | scottss 44267 | . . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
| 2 | 1 | sseli 3978 | . 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑}) | 
| 3 | vex 3483 | . . 3 ⊢ 𝑦 ∈ V | |
| 4 | elscottab.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 5 | 3, 4 | elab 3678 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) | 
| 6 | 2, 5 | sylib 218 | 1 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 {cab 2713 Scott cscott 44259 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-ss 3967 df-scott 44260 | 
| This theorem is referenced by: cpcolld 44282 grucollcld 44284 | 
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