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Mathbox for Rohan Ridenour |
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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 42615 | . . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
2 | 1 | sseli 3944 | . 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
3 | vex 3451 | . . 3 ⊢ 𝑦 ∈ V | |
4 | elscottab.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | elab 3634 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
6 | 2, 5 | sylib 217 | 1 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 {cab 2710 Scott cscott 42607 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-in 3921 df-ss 3931 df-scott 42608 |
This theorem is referenced by: cpcolld 42630 grucollcld 42632 |
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