Theorem elpwunsn 4575

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 3854	. 2 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2		elpwg 4492	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		dfss3 3866	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4	2, 3	bitrdi 290	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
5	4	notbid 321	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
6	5	biimpa 480	. . . 4 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7		rexnal 3151	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
8	6, 7	sylibr 237	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9		elpwi 4498	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10		ssel 3871	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})))
11		elun 4040	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}))
12		elsni 4534	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
13	12	orim2i 910	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐶))
14	13	ord 863	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶))
15	11, 14	sylbi 220	. . . . . . . . . . . 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 414	. . . . . . . . . 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 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
20	19	biimpd 232	. . . . . . . . 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 419	. . . . . . 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 3193	. . . 4 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))
26	25	imp 410	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴)
27	8, 26	syldan 594	. 2 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
28	1, 27	sylbi 220	1 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 846   = wceq 1542   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054   ∖ cdif 3841   ∪ cun 3842   ⊆ wss 3844  𝒫 cpw 4489  {csn 4517

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pw 4491  df-sn 4518

This theorem is referenced by:  pwfilem  8776  pwfilemOLD  8892  incexclem  15285  ramub1lem1  16463  ptcmplem5  22808  onsucsuccmpi  34270