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Theorem sspw 4535
Description: The powerclass preserves inclusion. See sspwb 5329 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5329 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
Assertion
Ref Expression
sspw (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3960 . . . 4 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 velpw 4527 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
4 velpw 4527 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
52, 3, 43imtr4g 299 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
65ssrdv 3959 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  wss 3919  𝒫 cpw 4522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524
This theorem is referenced by:  sspwi  4536  sspwd  4537  sspwb  5329  r1pwss  9210  elsigagen2  31464  measres  31538
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