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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 43785)
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 44134 is sspwimpVD 44135 without virtual deductions and was derived
from sspwimpVD 44135. (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 3470 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | vd01 43813 | . . . . . 6 ⊢ ( ⊤ ▶ 𝑥 ∈ V ) |
3 | idn1 43790 | . . . . . . 7 ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 ) | |
4 | idn1 43790 | . . . . . . . 8 ⊢ ( 𝑥 ∈ 𝒫 𝐴 ▶ 𝑥 ∈ 𝒫 𝐴 ) | |
5 | elpwi 4601 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
6 | 4, 5 | el1 43844 | . . . . . . 7 ⊢ ( 𝑥 ∈ 𝒫 𝐴 ▶ 𝑥 ⊆ 𝐴 ) |
7 | sstr 3982 | . . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵) | |
8 | 7 | ancoms 458 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵) |
9 | 3, 6, 8 | el12 43942 | . . . . . 6 ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 ) |
10 | 2, 9 | elpwgdedVD 44133 | . . . . . 6 ⊢ ( ( ⊤ , ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ) ▶ 𝑥 ∈ 𝒫 𝐵 ) |
11 | 2, 9, 10 | un0.1 43995 | . . . . 5 ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵 ) |
12 | 11 | int2 43822 | . . . 4 ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ) |
13 | 12 | gen11 43832 | . . 3 ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ) |
14 | dfss2 3960 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) | |
15 | 14 | biimpri 227 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
16 | 13, 15 | el1 43844 | . 2 ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 ) |
17 | 16 | in1 43787 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 𝒫 cpw 4594 ( wvhc2 43796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3947 df-ss 3957 df-pw 4596 df-vd1 43786 df-vhc2 43797 |
This theorem is referenced by: (None) |
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