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Theorem sspwimp 40743
 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. For the biconditional, see sspwb 5226. The proof sspwimp 40743, using conventional notation, was translated from virtual deduction form, sspwimpVD 40744, 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 3435 . . . . . . 7 𝑥 ∈ V
21a1i 11 . . . . . 6 (⊤ → 𝑥 ∈ V)
3 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
4 id 22 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
5 elpwi 4457 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5syl 17 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sstr 3892 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 459 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8syl2an 595 . . . . . 6 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
102, 9elpwgded 40389 . . . . . 6 ((⊤ ∧ (𝐴𝐵𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
112, 9, 10uun0.1 40603 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
1211ex 413 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312alrimiv 1903 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
14 dfss2 3872 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 229 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1716iin1 40397 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1518  ⊤wtru 1521   ∈ wcel 2079  Vcvv 3432   ⊆ wss 3854  𝒫 cpw 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-in 3861  df-ss 3869  df-pw 4449 This theorem is referenced by: (None)
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