Theorem elpwid 3682

Description: An element of a power class is a subclass. Deduction form of elpwi 3680. (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 3680	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2205   ⊆ wss 3213  𝒫 cpw 3671

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673

This theorem is referenced by:  fopwdom  7091  ssenen  7107  fival  7259  fiuni  7267  3nelsucpw1  7546  elnp1st2nd  7796  ixxssxr  10239  elfzoelz  10488  ballotfilem2  13153  ballotfilemfmpn  13159  restid2  13482  epttop  15004  neiss2  15056  blssm  15335  blin2  15346  cncfrss  15489  cncfrss2  15490  dvidsslem  15607  dvconstss  15612  plybss  15647  uhgrss  16119  upgrss  16143  upgr1een  16168  usgrss  16221  eupth2lemsfi  16522  pw1ndom3lem  16812  pwle2  16821