Theorem pwel 4196

Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)

Assertion

Ref	Expression
pwel	⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)

Proof of Theorem pwel

Step	Hyp	Ref	Expression
1		elssuni 3817	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
2		sspwb 4194	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
3	1, 2	sylib 121	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
4		pwexg 4159	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
5		elpwg 3567	. . 3 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵))
6	4, 5	syl 14	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵))
7	3, 6	mpbird 166	1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  𝒫 cpw 3559  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-uni 3790

This theorem is referenced by: (None)