Theorem elpwd 4311

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 4310	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
5	1, 4	mpbird 247	1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2139   ⊆ wss 3715  𝒫 cpw 4302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304

This theorem is referenced by:  reprval  30997  scutval  32217  bj-discrmoore  33372  dmvolss  40705  sge0xaddlem1  41153  ovnval2b  41272  ovnsubadd2lem  41365  vonvolmbllem  41380  vonvolmbl  41381  smfresal  41501  smfpimbor1lem1  41511  sprsymrelfvlem  42250