MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspw Structured version   Visualization version   GIF version

Theorem sspw 4575
Description: The powerclass preserves inclusion. See sspwb 5410 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5410 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 3955 . . . 4 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 velpw 4569 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
4 velpw 4569 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
52, 3, 43imtr4g 296 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
65ssrdv 3954 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3914  𝒫 cpw 4564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566
This theorem is referenced by:  sspwi  4576  sspwd  4577  sspwb  5410  r1pwss  9728  elsigagen2  32811  measres  32885
  Copyright terms: Public domain W3C validator