Theorem elpwid 3663

Description: An element of a power class is a subclass. Deduction form of elpwi 3661. (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 3661	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2202   ⊆ wss 3200  𝒫 cpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  fopwdom  7022  ssenen  7037  fival  7169  fiuni  7177  3nelsucpw1  7452  elnp1st2nd  7696  ixxssxr  10135  elfzoelz  10382  restid2  13333  epttop  14817  neiss2  14869  blssm  15148  blin2  15159  cncfrss  15302  cncfrss2  15303  dvidsslem  15420  dvconstss  15425  plybss  15460  uhgrss  15929  upgrss  15953  upgr1een  15978  usgrss  16031  eupth2lemsfi  16332  pw1ndom3lem  16609  pwle2  16620