Theorem elpwun 7702

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 7693	. . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)
6	2, 5	sylibr 233	. 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V)
7		elpwg 4563	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶)))
8		uncom 4113	. . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
9	8	sseq2i 3973	. . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵))
10		ssundif 4445	. . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
11	9, 10	bitri 274	. . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)
12		difexg 5284	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V)
13		elpwg 4563	. . . . 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 289	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
16	7, 15	bitrd 278	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵))
17	1, 6, 16	pm5.21nii 379	1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ⊆ wss 3910  𝒫 cpw 4560

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-pw 4562  df-sn 4587  df-pr 4589  df-uni 4866

This theorem is referenced by:  pwfilem  9120  pwfilemOLD  9289  elrfi  40995  dssmapnvod  42274