Theorem elpwd 4609

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
elpwd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elpwd.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
elpwd	⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwd

Step	Hyp	Ref	Expression
1		elpwd.2	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		elpwd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		elpwg 4606	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2107   ⊆ wss 3949  𝒫 cpw 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605

This theorem is referenced by:  sselpwd  5327  pwel  5380  frd  5636  f1opw2  7661  pwuncl  7757  naddunif  8692  f1opwfi  9356  ackbij1lem6  10220  ackbij1lem11  10225  mreacs  17602  sylow3lem3  19497  sylow3lem6  19500  cmpcov  22893  tgqtop  23216  filss  23357  scutval  27301  madecut  27377  cofcut1  27407  cutlt  27419  fnpreimac  31896  pcmplfin  32840  rspectopn  32847  zarclsint  32852  zarcmplem  32861  indval  33011  reprval  33622  bj-sselpwuni  35931  bj-discrmoore  35992  dmvolss  44701  sge0xaddlem1  45149  meadjuni  45173  ovnval2b  45268  ovnsubadd2lem  45361  vonvolmbllem  45376  vonvolmbl  45377  smfresal  45504  smfpimbor1lem1  45514  sprsymrelfvlem  46158  lindslinindsimp1  47138  lindslinindimp2lem4  47142  lincresunit3  47162  iscnrm3llem1  47582