Theorem sspwi 4541

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 4540	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

Syntax hints:   ⊆ wss 3883  𝒫 cpw 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-pw 4531

This theorem is referenced by:  pwunss  4547  pwundif  4553  pwdom  9057  wdompwdom  9483  rankxplim  9794  hashbclem  14405  incexclem  15792  sscpwex  17773  wunfunc  17859  tsmsres  24127  cfilresi  25280  vitali  25598  sqff1o  27163  ldgenpisyslem1  34347  imambfm  34446  ballotlem2  34673  ttcpwss  36743  dssmapnvod  44464  gneispace  44578  sge0less  46835