Theorem sspwi 4579

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 4578	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

Syntax hints:   ⊆ wss 3913  𝒫 cpw 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-pw 4569

This theorem is referenced by:  pwunss  4585  pwundif  4592  pwdom  9116  wdompwdom  9539  rankxplim  9850  hashbclem  14488  incexclem  15889  sscpwex  17871  wunfunc  17957  tsmsres  24269  cfilresi  25422  vitali  25740  sqff1o  27311  ldgenpisyslem1  34497  imambfm  34596  ballotlem2  34823  ttcpwss  36914  dssmapnvod  44637  gneispace  44751  sge0less  46997