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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 40654 | . . 3 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
2 | 1 | sseli 3956 | . 2 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
3 | vex 3494 | . . 3 ⊢ 𝑦 ∈ V | |
4 | elscottab.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | elab 3663 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
6 | 2, 5 | sylib 220 | 1 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2113 {cab 2798 Scott cscott 40646 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-scott 40647 |
This theorem is referenced by: cpcolld 40669 grucollcld 40671 |
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