Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssabf | Structured version Visualization version GIF version |
Description: Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ssabf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
ssabf | ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssabf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | abid2f 3009 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
3 | 2 | sseq1i 3992 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ⊆ {𝑥 ∣ 𝜑} ↔ 𝐴 ⊆ {𝑥 ∣ 𝜑}) |
4 | ss2ab 4036 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
5 | 3, 4 | bitr3i 278 | 1 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-in 3940 df-ss 3949 |
This theorem is referenced by: ssrabf 41258 |
Copyright terms: Public domain | W3C validator |