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  9445  pwwf  9821  rankpwi  9837  ackbij2lem1  10232  fictb  10258  ssfin2  10334  ssfin3ds  10344  ttukeylem2  10524  hashbcss  17024  isacs1i  17669  mreacs  17670  acsfn  17671  isacs3lem  18552  isacs5lem  18555  tgcmp  23339  imastopn  23658  fgabs  23817  fgtr  23828  trfg  23829  ssufl  23856  alexsubb  23984  cfiluweak  24233  cmetss  25268  minveclem4a  25382  minveclem4  25384  madess  27840  ldsysgenld  34191  neibastop1  36377  neibastop2lem  36378  neibastop2  36379  sstotbnd2  37798  prjcrv0  42656  isnacs3  42733  aomclem2  43079  sge0iunmptlemre  46444