Theorem elpwid 3679

Description: An element of a power class is a subclass. Deduction form of elpwi 3677. (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 3677	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2203   ⊆ wss 3210  𝒫 cpw 3668

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670

This theorem is referenced by:  fopwdom  7088  ssenen  7104  fival  7256  fiuni  7264  3nelsucpw1  7543  elnp1st2nd  7787  ixxssxr  10229  elfzoelz  10477  restid2  13450  epttop  14942  neiss2  14994  blssm  15273  blin2  15284  cncfrss  15427  cncfrss2  15428  dvidsslem  15545  dvconstss  15550  plybss  15585  uhgrss  16057  upgrss  16081  upgr1een  16106  usgrss  16159  eupth2lemsfi  16460  pw1ndom3lem  16750  pwle2  16759