Theorem sspwi 4576

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 4575	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

This proof depends on syntax axioms:   ⊆ wss 3906  𝒫 cpw 4564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-pw 4566

This theorem is used by:  pwunss  4582  pwundif  4589  pwdom  9124  wdompwdom  9547  rankxplim  9858  hashbclem  14507  incexclem  15913  sscpwex  17894  wunfunc  17980  tsmsres  24352  cfilresi  25505  vitali  25823  sqff1o  27397  ldgenpisyslem1  34618  imambfm  34717  ballotlem2  34944  ttcpwss  37083  dssmapnvod  44804  gneispace  44918  sge0less  47164