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Mirrors > Home > ILE Home > Th. List > ssrab | GIF version |
Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.) |
Ref | Expression |
---|---|
ssrab | ⊢ (B ⊆ {x ∈ A ∣ φ} ↔ (B ⊆ A ∧ ∀x ∈ B φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2309 | . . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
2 | 1 | sseq2i 2964 | . 2 ⊢ (B ⊆ {x ∈ A ∣ φ} ↔ B ⊆ {x ∣ (x ∈ A ∧ φ)}) |
3 | ssab 3004 | . 2 ⊢ (B ⊆ {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ B → (x ∈ A ∧ φ))) | |
4 | dfss3 2929 | . . . 4 ⊢ (B ⊆ A ↔ ∀x ∈ B x ∈ A) | |
5 | 4 | anbi1i 431 | . . 3 ⊢ ((B ⊆ A ∧ ∀x ∈ B φ) ↔ (∀x ∈ B x ∈ A ∧ ∀x ∈ B φ)) |
6 | r19.26 2435 | . . 3 ⊢ (∀x ∈ B (x ∈ A ∧ φ) ↔ (∀x ∈ B x ∈ A ∧ ∀x ∈ B φ)) | |
7 | df-ral 2305 | . . 3 ⊢ (∀x ∈ B (x ∈ A ∧ φ) ↔ ∀x(x ∈ B → (x ∈ A ∧ φ))) | |
8 | 5, 6, 7 | 3bitr2ri 198 | . 2 ⊢ (∀x(x ∈ B → (x ∈ A ∧ φ)) ↔ (B ⊆ A ∧ ∀x ∈ B φ)) |
9 | 2, 3, 8 | 3bitri 195 | 1 ⊢ (B ⊆ {x ∈ A ∣ φ} ↔ (B ⊆ A ∧ ∀x ∈ B φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 {cab 2023 ∀wral 2300 {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-ral 2305 df-rab 2309 df-in 2918 df-ss 2925 |
This theorem is referenced by: ssrabdv 3013 |
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