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Mirrors > Home > NFE Home > Th. List > rabss2 | GIF version |
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rabss2 | ⊢ (A ⊆ B → {x ∈ A ∣ φ} ⊆ {x ∈ B ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.45 807 | . . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ∧ φ) → (x ∈ B ∧ φ))) | |
2 | 1 | alimi 1559 | . . 3 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ φ))) |
3 | dfss2 3262 | . . 3 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
4 | ss2ab 3334 | . . 3 ⊢ ({x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ (x ∈ B ∧ φ)} ↔ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ φ))) | |
5 | 2, 3, 4 | 3imtr4i 257 | . 2 ⊢ (A ⊆ B → {x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ (x ∈ B ∧ φ)}) |
6 | df-rab 2623 | . 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
7 | df-rab 2623 | . 2 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)} | |
8 | 5, 6, 7 | 3sstr4g 3312 | 1 ⊢ (A ⊆ B → {x ∈ A ∣ φ} ⊆ {x ∈ B ∣ φ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 {cab 2339 {crab 2618 ⊆ wss 3257 |
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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: (None) |
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