Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sspwimp Structured version   Visualization version   GIF version

Theorem sspwimp 38176
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimp 38176, using conventional notation, was translated from virtual deduction form, sspwimpVD 38177, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimp (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . . . 7 𝑥 ∈ V
21a1i 11 . . . . . 6 (⊤ → 𝑥 ∈ V)
3 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
4 id 22 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
5 elpwi 4117 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5syl 17 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sstr 3576 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 468 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8syl2an 493 . . . . . 6 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
102, 9elpwgded 37801 . . . . . 6 ((⊤ ∧ (𝐴𝐵𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
112, 9, 10uun0.1 38026 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
1211ex 449 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312alrimiv 1842 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
14 dfss2 3557 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 217 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1716iin1 37809 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wtru 1476  wcel 1977  Vcvv 3173  wss 3540  𝒫 cpw 4108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator