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Theorem sspwimpcfVD 39471
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 39102) 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 39470 is sspwimpcfVD 39471 without virtual deductions and was derived from sspwimpcfVD 39471. 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 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpcfVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . . . . 6 𝑥 ∈ V
2 idn1 39107 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn1 39107 . . . . . . . 8 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
4 elpwi 4201 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4el1 39170 . . . . . . 7 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥𝐴   )
6 sstr2 3643 . . . . . . . 8 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
76impcom 445 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
82, 5, 7el12 39270 . . . . . 6 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
9 elpwg 4199 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
109biimpar 501 . . . . . 6 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
111, 8, 10el021old 39243 . . . . 5 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
1211int2 39148 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
1312gen11 39158 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
14 dfss2 3624 . . . 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 39170 . 2 (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
1716in1 39104 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-vd1 39103  df-vhc2 39114 This theorem is referenced by: (None)
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