Theorem sspwi 4575

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 4574	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	ax-mp 5	1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵

Syntax hints:   ⊆ wss 3914  𝒫 cpw 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-ss 3931  df-pw 4565

This theorem is referenced by:  pwunss  4581  pwundif  4587  pwdom  9093  wdompwdom  9531  rankxplim  9832  hashbclem  14417  incexclem  15802  sscpwex  17777  wunfunc  17863  tsmsres  24031  cfilresi  25195  vitali  25514  sqff1o  27092  ldgenpisyslem1  34153  imambfm  34253  ballotlem2  34480  dssmapnvod  44009  gneispace  44123  sge0less  46390