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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 44239 | . . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
2 | 1 | sseli 3991 | . 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
3 | vex 3482 | . . 3 ⊢ 𝑦 ∈ V | |
4 | elscottab.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | elab 3681 | . 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 2106 {cab 2712 Scott cscott 44231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-ss 3980 df-scott 44232 |
This theorem is referenced by: cpcolld 44254 grucollcld 44256 |
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