Theorem elpwunsn 4619

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

Assertion

Ref	Expression
elpwunsn	⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)

Proof of Theorem elpwunsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3897	. 2 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2		elpwg 4536	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		dfss3 3909	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4	2, 3	bitrdi 287	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
5	4	notbid 318	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
6	5	biimpa 477	. . . 4 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7		rexnal 3169	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
8	6, 7	sylibr 233	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9		elpwi 4542	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10		ssel 3914	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})))
11		elun 4083	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}))
12		elsni 4578	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
13	12	orim2i 908	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐶))
14	13	ord 861	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶))
15	11, 14	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐵 ∪ {𝐶}) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶))
16	15	imim2i 16	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶)))
17	16	impd 411	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 = 𝐶))
18	9, 10, 17	3syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 = 𝐶))
19		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
20	19	biimpd 228	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴))
21	18, 20	syl6 35	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴)))
22	21	expd 416	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴))))
23	22	com4r 94	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))))
24	23	pm2.43b 55	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴)))
25	24	rexlimdv 3212	. . . 4 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))
26	25	imp 407	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴)
27	8, 26	syldan 591	. 2 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
28	1, 27	sylbi 216	1 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pw 4535  df-sn 4562

This theorem is referenced by:  pwfilem  8960  pwfilemOLD  9113  incexclem  15548  ramub1lem1  16727  ptcmplem5  23207  onsucsuccmpi  34632