Theorem elpwunsn 4645

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 3909	. 2 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2		elpwg 4560	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		dfss3 3920	. . . . . . 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 482	. . . 4 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7		rexnal 3115	. . . 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 4564	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10		ssel 3925	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})))
11		elun 4100	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}))
12		elsni 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
13	12	orim2i 924	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐶))
14	13	ord 878	. . . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶)))
17	16	impd 416	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 = 𝐶))
18	9, 10, 17	3syl 19	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → 𝑥 = 𝐶))
19		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
20	19	biimpd 232	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴))
21	18, 20	syl6 36	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴)))
22	21	expd 421	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐴))))
23	22	com4r 95	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))))
24	23	pm2.43b 56	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴)))
25	24	rexlimdv 3162	. . . 4 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))
26	25	imp 412	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴)
27	8, 26	syldan 603	. 2 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
28	1, 27	sylbi 220	1 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-pw 4559  df-sn 4585

This theorem is used by:  pwfilem  9293  incexclem  15985  ramub1lem1  17184  ptcmplem5  24355  onsucsuccmpi  37201