Theorem pwel 4203

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 3824	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)
2		sspwb 4201	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
3	1, 2	sylib 121	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵)
4		pwexg 4166	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
5		elpwg 3574	. . 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 2141  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)