Theorem elpwid 3640

Description: An element of a power class is a subclass. Deduction form of elpwi 3638. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
elpwid.1	⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
elpwid	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem elpwid

Step	Hyp	Ref	Expression
1		elpwid.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
2		elpwi 3638	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2180   ⊆ wss 3177  𝒫 cpw 3629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631

This theorem is referenced by:  fopwdom  6965  ssenen  6980  fival  7105  fiuni  7113  3nelsucpw1  7387  elnp1st2nd  7631  ixxssxr  10064  elfzoelz  10311  restid2  13247  epttop  14729  neiss2  14781  blssm  15060  blin2  15071  cncfrss  15214  cncfrss2  15215  dvidsslem  15332  dvconstss  15337  plybss  15372  uhgrss  15840  upgrss  15864  usgrss  15940  pwle2  16275