![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rabssab | GIF version |
Description: A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabssab | ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2309 | . 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
2 | simpr 103 | . . 3 ⊢ ((x ∈ A ∧ φ) → φ) | |
3 | 2 | ss2abi 3006 | . 2 ⊢ {x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ φ} |
4 | 1, 3 | eqsstri 2969 | 1 ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1390 {cab 2023 {crab 2304 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-in 2918 df-ss 2925 |
This theorem is referenced by: epse 4064 riotasbc 5426 genipv 6492 |
Copyright terms: Public domain | W3C validator |