Theorem elpwd 4549

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 4544	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
5	1, 4	mpbird 259	1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2110   ⊆ wss 3935  𝒫 cpw 4538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-in 3942  df-ss 3951  df-pw 4540

This theorem is referenced by:  sselpwd  5222  pwel  5336  f1opw2  7394  pwuncl  7486  f1opwfi  8822  ackbij1lem6  9641  ackbij1lem11  9646  mreacs  16923  sylow3lem3  18748  sylow3lem6  18751  cmpcov  21991  tgqtop  22314  filss  22455  fnpreimac  30410  pcmplfin  31119  indval  31267  reprval  31876  scutval  33260  bj-sselpwuni  34337  bj-discrmoore  34397  dmvolss  42264  sge0xaddlem1  42709  meadjuni  42733  ovnval2b  42828  ovnsubadd2lem  42921  vonvolmbllem  42936  vonvolmbl  42937  smfresal  43057  smfpimbor1lem1  43067  sprsymrelfvlem  43646  lindslinindsimp1  44506  lindslinindimp2lem4  44510  lincresunit3  44530