Theorem sspwd 4613

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 4611	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3951  𝒫 cpw 4600

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-pw 4602

This theorem is referenced by:  pweq  4614  pwel  5381  marypha1lem  9473  pwwf  9847  rankpwi  9863  ackbij2lem1  10258  fictb  10284  ssfin2  10360  ssfin3ds  10370  ttukeylem2  10550  hashbcss  17042  isacs1i  17700  mreacs  17701  acsfn  17702  isacs3lem  18587  isacs5lem  18590  tgcmp  23409  imastopn  23728  fgabs  23887  fgtr  23898  trfg  23899  ssufl  23926  alexsubb  24054  cfiluweak  24304  cmetss  25350  minveclem4a  25464  minveclem4  25466  madess  27915  ldsysgenld  34161  neibastop1  36360  neibastop2lem  36361  neibastop2  36362  sstotbnd2  37781  prjcrv0  42643  isnacs3  42721  aomclem2  43067  sge0iunmptlemre  46430