Theorem elpwun 7211

Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)

Hypothesis

Ref	Expression
eldifpw.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elpwun	⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)

Proof of Theorem elpwun

Step	Hyp	Ref	Expression
1		elex 3400	. 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V)
2		elex 3400	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V)
3		eldifpw.1	. . . 4 ⊢ 𝐶 ∈ V
4		difex2 7202	. . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)
6	2, 5	sylibr 226	. 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V)
7		elpwg 4357	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶)))
8		difexg 5003	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V)
9		elpwg 4357	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵))
10	8, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵))
11		uncom 3955	. . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
12	11	sseq2i 3826	. . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵))
13		ssundif 4246	. . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
14	12, 13	bitri 267	. . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
15	10, 14	syl6rbbr 282	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
16	7, 15	bitrd 271	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
17	1, 6, 16	pm5.21nii 370	1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 198   ∈ wcel 2157  Vcvv 3385   ∖ cdif 3766   ∪ cun 3767   ⊆ wss 3769  𝒫 cpw 4349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-sn 4369  df-pr 4371  df-uni 4629

This theorem is referenced by:  pwfilem  8502  elrfi  38043  dssmapnvod  39096