Theorem eldifpw 7612

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 4548	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		unss1 4118	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
3		eldifpw.1	. . . . . . 7 ⊢ 𝐶 ∈ V
4		unexg 7593	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
5	3, 4	mpan2 688	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ V)
6		elpwg 4542	. . . . . 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	syl5ibr 245	. . . 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 4548	. . . . 5 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ⊆ 𝐵)
11	10	unssbd 4127	. . . 4 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → 𝐶 ⊆ 𝐵)
12	11	con3i 154	. . 3 ⊢ (¬ 𝐶 ⊆ 𝐵 → ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)
13	9, 12	anim12i 613	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
14		eldif 3902	. 2 ⊢ ((𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
15	13, 14	sylibr 233	1 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2110  Vcvv 3431   ∖ cdif 3889   ∪ cun 3890   ⊆ wss 3892  𝒫 cpw 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-pw 4541  df-sn 4568  df-pr 4570  df-uni 4846

This theorem is referenced by: (None)