Theorem sspwimpcfVD 44892

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44540) 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 44891 is sspwimpcfVD 44892 without virtual deductions and was derived from sspwimpcfVD 44892. 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 3492	. . . . . 6 ⊢ 𝑥 ∈ V
2		idn1 44545	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝐴 ⊆ 𝐵   )
3		idn1 44545	. . . . . . . 8 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
4		elpwi 4629	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
5	3, 4	el1 44599	. . . . . . 7 ⊢ (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		sstr2 4015	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
7	6	impcom 407	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐵)
8	2, 5, 7	el12 44697	. . . . . 6 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ⊆ 𝐵   )
9		elpwg 4625	. . . . . . 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 44672	. . . . 5 ⊢ (   (   𝐴 ⊆ 𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
12	11	int2 44577	. . . 4 ⊢ (   𝐴 ⊆ 𝐵   ▶   (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
13	12	gen11 44587	. . 3 ⊢ (   𝐴 ⊆ 𝐵   ▶   ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)   )
14		df-ss 3993	. . . 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 44599	. 2 ⊢ (   𝐴 ⊆ 𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
17	16	in1 44542	1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-pw 4624  df-vd1 44541  df-vhc2 44552

This theorem is referenced by: (None)