Theorem sspwimpVD 42428

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 42078) 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 42427 is sspwimpVD 42428 without virtual deductions and was derived from sspwimpVD 42428. (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 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	vd01 42106	. . . . . 6 ⊢ (   ⊤   ▶   𝑥 ∈ V   )
3		idn1 42083	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   )
4		idn1 42083	. . . . . . . 8 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
5		elpwi 4539	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
6	4, 5	el1 42137	. . . . . . 7 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ⊆ 𝐴   )
7		sstr 3925	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
8	7	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
9	3, 6, 8	el12 42235	. . . . . 6 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   )
10	2, 9	elpwgdedVD 42426	. . . . . 6 ⊢ (   (   ⊤   ,   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   )   ▶   𝑥 ∈ 𝒫 𝐵   )
11	2, 9, 10	un0.1 42288	. . . . 5 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
12	11	int2 42115	. . . 4 ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
13	12	gen11 42125	. . 3 ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
14		dfss2 3903	. . . 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 42137	. 2 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
17	16	in1 42080	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1537  ⊤wtru 1540   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  (   wvhc2 42089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-vd1 42079  df-vhc2 42090

This theorem is referenced by: (None)