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Theorem sspwimpVD 41618
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 41268) 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 41617 is sspwimpVD 41618 without virtual deductions and was derived from sspwimpVD 41618. (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 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3447 . . . . . . 7 𝑥 ∈ V
21vd01 41296 . . . . . 6 (      ▶   𝑥 ∈ V   )
3 idn1 41273 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
4 idn1 41273 . . . . . . . 8 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
5 elpwi 4509 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5el1 41327 . . . . . . 7 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥𝐴   )
7 sstr 3926 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 462 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8el12 41425 . . . . . 6 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
102, 9elpwgdedVD 41616 . . . . . 6 (   (      ,   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   )   ▶   𝑥 ∈ 𝒫 𝐵   )
112, 9, 10un0.1 41478 . . . . 5 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
1211int2 41305 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
1312gen11 41315 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
14 dfss2 3904 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 231 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15el1 41327 . 2 (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
1716in1 41270 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ⊤wtru 1539   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  𝒫 cpw 4500  (   wvhc2 41279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-vd1 41269  df-vhc2 41280 This theorem is referenced by: (None)
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