Theorem sspwd 4588

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

Hypothesis

Ref	Expression
sspwd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
sspwd	⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwd

Step	Hyp	Ref	Expression
1		sspwd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sspw 4586	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3926  𝒫 cpw 4575

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-ss 3943  df-pw 4577

This theorem is referenced by:  pweq  4589  pwel  5351  marypha1lem  9443  pwwf  9819  rankpwi  9835  ackbij2lem1  10230  fictb  10256  ssfin2  10332  ssfin3ds  10342  ttukeylem2  10522  hashbcss  17022  isacs1i  17667  mreacs  17668  acsfn  17669  isacs3lem  18550  isacs5lem  18553  tgcmp  23337  imastopn  23656  fgabs  23815  fgtr  23826  trfg  23827  ssufl  23854  alexsubb  23982  cfiluweak  24231  cmetss  25266  minveclem4a  25380  minveclem4  25382  madess  27832  ldsysgenld  34137  neibastop1  36323  neibastop2lem  36324  neibastop2  36325  sstotbnd2  37744  prjcrv0  42603  isnacs3  42680  aomclem2  43026  sge0iunmptlemre  46392