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Mathbox for Alan Sare |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwimpVD | Structured version Visualization version GIF version |
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44566)
using conjunction-form virtual hypothesis collections. It was completed
manually, but has the potential to be completed automatically by a tools
program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant.
sspwimp 44915 is sspwimpVD 44916 without virtual deductions and was derived
from sspwimpVD 44916. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
Ref | Expression |
---|---|
sspwimpVD | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | vd01 44594 | . . . . . 6 ⊢ ( ⊤ ▶ 𝑥 ∈ V ) |
3 | idn1 44571 | . . . . . . 7 ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 ) | |
4 | idn1 44571 | . . . . . . . 8 ⊢ ( 𝑥 ∈ 𝒫 𝐴 ▶ 𝑥 ∈ 𝒫 𝐴 ) | |
5 | elpwi 4611 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
6 | 4, 5 | el1 44625 | . . . . . . 7 ⊢ ( 𝑥 ∈ 𝒫 𝐴 ▶ 𝑥 ⊆ 𝐴 ) |
7 | sstr 4003 | . . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵) | |
8 | 7 | ancoms 458 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵) |
9 | 3, 6, 8 | el12 44723 | . . . . . 6 ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 ) |
10 | 2, 9 | elpwgdedVD 44914 | . . . . . 6 ⊢ ( ( ⊤ , ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ) ▶ 𝑥 ∈ 𝒫 𝐵 ) |
11 | 2, 9, 10 | un0.1 44776 | . . . . 5 ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵 ) |
12 | 11 | int2 44603 | . . . 4 ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ) |
13 | 12 | gen11 44613 | . . 3 ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ) |
14 | df-ss 3979 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) | |
15 | 14 | biimpri 228 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
16 | 13, 15 | el1 44625 | . 2 ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |
17 | 16 | in1 44568 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ⊤wtru 1537 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 ( wvhc2 44577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-ss 3979 df-pw 4606 df-vd1 44567 df-vhc2 44578 |
This theorem is referenced by: (None) |
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