Theorem sspwimpcfVD 45161

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44810) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf 45160 is sspwimpcfVD 45161 without virtual deductions and was derived from sspwimpcfVD 45161. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-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
sspwimpcfVD	⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpcfVD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
2		idn1 44815	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   )
3		idn1 44815	. . . . . . . 8 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
4		elpwi 4561	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
5	3, 4	el1 44869	. . . . . . 7 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		sstr2 3940	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
7	6	impcom 407	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
8	2, 5, 7	el12 44966	. . . . . 6 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   )
9		elpwg 4557	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵))
10	9	biimpar 477	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ 𝒫 𝐵)
11	1, 8, 10	el021old 44942	. . . . 5 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
12	11	int2 44847	. . . 4 ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
13	12	gen11 44857	. . 3 ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
14		df-ss 3918	. . . 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 44869	. 2 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
17	16	in1 44812	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-ss 3918  df-pw 4556  df-vd1 44811  df-vhc2 44822

This theorem is referenced by: (None)