Theorem sspwimpVD 44908

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44559) 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 44907 is sspwimpVD 44908 without virtual deductions and was derived from sspwimpVD 44908. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   )
2::	⊢ (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   )
3:2:	⊢ (   .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ⊆ 𝐴   )
4:3,1:	⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   )
5::	⊢ 𝑥 ∈ V
6:4,5:	⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    )
7:6:	⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)    )
8:7:	⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
9:8:	⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
qed:9:	⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Assertion

Ref	Expression
sspwimpVD	⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpVD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	vd01 44587	. . . . . 6 ⊢ (   ⊤   ▶   𝑥 ∈ V   )
3		idn1 44564	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   )
4		idn1 44564	. . . . . . . 8 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
5		elpwi 4570	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6	4, 5	el1 44618	. . . . . . 7 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ⊆ 𝐴   )
7		sstr 3955	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
8	7	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
9	3, 6, 8	el12 44715	. . . . . 6 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   )
10	2, 9	elpwgdedVD 44906	. . . . . 6 ⊢ (   (   ⊤   ,   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   )   ▶   𝑥 ∈ 𝒫 𝐵   )
11	2, 9, 10	un0.1 44768	. . . . 5 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
12	11	int2 44596	. . . 4 ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
13	12	gen11 44606	. . 3 ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
14		df-ss 3931	. . . 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 44618	. 2 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
17	16	in1 44561	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1538  ⊤wtru 1541   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  (   wvhc2 44570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-ss 3931  df-pw 4565  df-vd1 44560  df-vhc2 44571

This theorem is referenced by: (None)