Theorem elpwi 3658

Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elpwi	⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)

Proof of Theorem elpwi

Step	Hyp	Ref	Expression
1		elpwg 3657	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
2	1	ibi 176	1 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2200   ⊆ wss 3197  𝒫 cpw 3649

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  elpwid  3660  elelpwi  3661  elpw2g  4239  eldifpw  4567  iunpw  4570  f1opw2  6210  pw1dc1  7072  fi0  7138  pw1m  7405  pw1on  7407  lspf  14347  cnntr  14893  edgssv2en  15991  2omap  16318  pw1map  16320