Theorem sspwi 4553

Description: The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)

Hypothesis

Ref	Expression
sspwi.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
sspwi	⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

Proof of Theorem sspwi

Step	Hyp	Ref	Expression
1		sspwi.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		sspw 4552	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

Syntax hints:   ⊆ wss 3889  𝒫 cpw 4541

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-pw 4543

This theorem is referenced by:  pwunss  4559  pwundif  4565  pwdom  9067  wdompwdom  9493  rankxplim  9803  hashbclem  14414  incexclem  15801  sscpwex  17782  wunfunc  17868  tsmsres  24109  cfilresi  25262  vitali  25580  sqff1o  27145  ldgenpisyslem1  34307  imambfm  34406  ballotlem2  34633  ttcpwss  36697  dssmapnvod  44447  gneispace  44561  sge0less  46820