Theorem elpwun 7724

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 3463	. 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V)
2		elex 3463	. . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V)
3		eldifpw.1	. . . 4 ⊢ 𝐶 ∈ V
4		difex2 7715	. . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)
6	2, 5	sylibr 234	. 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V)
7		elpwg 4559	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶)))
8		uncom 4112	. . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
9	8	sseq2i 3965	. . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵))
10		ssundif 4442	. . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
11	9, 10	bitri 275	. . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
12		difexg 5276	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V)
13		elpwg 4559	. . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵))
14	12, 13	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵))
15	11, 14	bitr4id 290	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
16	7, 15	bitrd 279	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
17	1, 6, 16	pm5.21nii 378	1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  𝒫 cpw 4556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-pw 4558  df-sn 4583  df-pr 4585  df-uni 4866

This theorem is referenced by:  pwfilem  9230  elrfi  43045  dssmapnvod  44370