Theorem eldifpw 7803

Description: Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)

Hypothesis

Ref	Expression
eldifpw.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
eldifpw	⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Proof of Theorem eldifpw

Step	Hyp	Ref	Expression
1		elpwi 4629	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		unss1 4208	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
3		eldifpw.1	. . . . . . 7 ⊢ 𝐶 ∈ V
4		unexg 7778	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
5	3, 4	mpan2 690	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ V)
6		elpwg 4625	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) ∈ V → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝐵 → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
8	2, 7	imbitrrid 246	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶)))
9	1, 8	mpd 15	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶))
10		elpwi 4629	. . . . 5 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ⊆ 𝐵)
11	10	unssbd 4217	. . . 4 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → 𝐶 ⊆ 𝐵)
12	11	con3i 154	. . 3 ⊢ (¬ 𝐶 ⊆ 𝐵 → ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)
13	9, 12	anim12i 612	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
14		eldif 3986	. 2 ⊢ ((𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
15	13, 14	sylibr 234	1 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932

This theorem is referenced by: (None)