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Theorem sspwimpVD 39654
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 39287) 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 39653 is sspwimpVD 39654 without virtual deductions and was derived from sspwimpVD 39654. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (    .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥 ∈ 𝒫 𝐴   )
3:2: (    .............. 𝑥 ∈ 𝒫 𝐴    ▶   𝑥𝐴   )
4:3,1: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
5:: 𝑥 ∈ V
6:4,5: (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵    )
7:6: (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)    )
8:7: (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 𝒫 𝐵)   )
9:8: (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
qed:9: (𝐴𝐵 → 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 3343 . . . . . . 7 𝑥 ∈ V
21vd01 39324 . . . . . 6 (      ▶   𝑥 ∈ V   )
3 idn1 39292 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
4 idn1 39292 . . . . . . . 8 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
5 elpwi 4312 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5el1 39355 . . . . . . 7 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥𝐴   )
7 sstr 3752 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 468 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8el12 39455 . . . . . 6 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
102, 9elpwgdedVD 39652 . . . . . 6 (   (      ,   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   )   ▶   𝑥 ∈ 𝒫 𝐵   )
112, 9, 10un0.1 39508 . . . . 5 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
1211int2 39333 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
1312gen11 39343 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
14 dfss2 3732 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 218 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15el1 39355 . 2 (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
1716in1 39289 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wtru 1633  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302  (   wvhc2 39298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-vd1 39288  df-vhc2 39299
This theorem is referenced by: (None)
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