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Mirrors > Home > NFE Home > Th. List > ssintrab | GIF version |
Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.) |
Ref | Expression |
---|---|
ssintrab | ⊢ (A ⊆ ∩{x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ⊆ x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2623 | . . . 4 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)} | |
2 | 1 | inteqi 3930 | . . 3 ⊢ ∩{x ∈ B ∣ φ} = ∩{x ∣ (x ∈ B ∧ φ)} |
3 | 2 | sseq2i 3296 | . 2 ⊢ (A ⊆ ∩{x ∈ B ∣ φ} ↔ A ⊆ ∩{x ∣ (x ∈ B ∧ φ)}) |
4 | impexp 433 | . . . 4 ⊢ (((x ∈ B ∧ φ) → A ⊆ x) ↔ (x ∈ B → (φ → A ⊆ x))) | |
5 | 4 | albii 1566 | . . 3 ⊢ (∀x((x ∈ B ∧ φ) → A ⊆ x) ↔ ∀x(x ∈ B → (φ → A ⊆ x))) |
6 | ssintab 3943 | . . 3 ⊢ (A ⊆ ∩{x ∣ (x ∈ B ∧ φ)} ↔ ∀x((x ∈ B ∧ φ) → A ⊆ x)) | |
7 | df-ral 2619 | . . 3 ⊢ (∀x ∈ B (φ → A ⊆ x) ↔ ∀x(x ∈ B → (φ → A ⊆ x))) | |
8 | 5, 6, 7 | 3bitr4i 268 | . 2 ⊢ (A ⊆ ∩{x ∣ (x ∈ B ∧ φ)} ↔ ∀x ∈ B (φ → A ⊆ x)) |
9 | 3, 8 | bitri 240 | 1 ⊢ (A ⊆ ∩{x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ⊆ x)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 {cab 2339 ∀wral 2614 {crab 2618 ⊆ wss 3257 ∩cint 3926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-int 3927 |
This theorem is referenced by: (None) |
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