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Theorem sspwimpVD 40706
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 40356) 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 40705 is sspwimpVD 40706 without virtual deductions and was derived from sspwimpVD 40706. (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 3413 . . . . . . 7 𝑥 ∈ V
21vd01 40384 . . . . . 6 (      ▶   𝑥 ∈ V   )
3 idn1 40361 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
4 idn1 40361 . . . . . . . 8 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
5 elpwi 4427 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5el1 40415 . . . . . . 7 (   𝑥 ∈ 𝒫 𝐴   ▶   𝑥𝐴   )
7 sstr 3861 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 451 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8el12 40513 . . . . . 6 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥𝐵   )
102, 9elpwgdedVD 40704 . . . . . 6 (   (      ,   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   )   ▶   𝑥 ∈ 𝒫 𝐵   )
112, 9, 10un0.1 40566 . . . . 5 (   (   𝐴𝐵   ,   𝑥 ∈ 𝒫 𝐴   )   ▶   𝑥 ∈ 𝒫 𝐵   )
1211int2 40393 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
1312gen11 40403 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵)   )
14 dfss2 3841 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 220 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15el1 40415 . 2 (   𝐴𝐵   ▶   𝒫 𝐴 ⊆ 𝒫 𝐵   )
1716in1 40358 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1506  ⊤wtru 1509   ∈ wcel 2051  Vcvv 3410   ⊆ wss 3824  𝒫 cpw 4417  (   wvhc2 40367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-v 3412  df-in 3831  df-ss 3838  df-pw 4419  df-vd1 40357  df-vhc2 40368 This theorem is referenced by: (None)
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