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Theorem sspwimpcfVD 39809
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 39447) 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 39808 is sspwimpcfVD 39809 without virtual deductions and was derived from sspwimpcfVD 39809. 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:: (   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
sspwimpcfVD (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpcfVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3353 . . . . . 6 𝑥 ∈ V
2 idn1 39452 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn1 39452 . . . . . . . 8 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
4 elpwi 4325 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4el1 39515 . . . . . . 7 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥𝐴   )
6 sstr2 3768 . . . . . . . 8 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
76impcom 396 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
82, 5, 7el12 39614 . . . . . 6 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
9 elpwg 4323 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
109biimpar 469 . . . . . 6 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
111, 8, 10el021old 39588 . . . . 5 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
1211int2 39493 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
1312gen11 39503 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
14 dfss2 3749 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 219 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15el1 39515 . 2 (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
1716in1 39449 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wcel 2155  Vcvv 3350  wss 3732  𝒫 cpw 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3739  df-ss 3746  df-pw 4317  df-vd1 39448  df-vhc2 39459
This theorem is referenced by: (None)
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