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 3895	. 2 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2		elpwg 4535	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		dfss3 3906	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4	2, 3	bitrdi 289	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
5	4	notbid 320	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵))
6	5	biimpa 478	. . . 4 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7		rexnal 3093	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
8	6, 7	sylibr 236	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9		elpwi 4539	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10		ssel 3911	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∪ {𝐶})))
11		elun 4086	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}))
12		elsni 4575	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
13	12	orim2i 917	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐶))
14	13	ord 871	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ {𝐶}) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐶))
15	11, 14	sylbi 219	. . . . . . . . . . . 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 412	. . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
20	19	biimpd 231	. . . . . . . . 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 417	. . . . . . 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 3140	. . . 4 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐶 ∈ 𝐴))
26	25	imp 408	. . 3 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴)
27	8, 26	syldan 598	. 2 ⊢ ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
28	1, 27	sylbi 219	1 ⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ∖ cdif 3882   ∪ cun 3883   ⊆ wss 3885  𝒫 cpw 4532  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-pw 4534  df-sn 4559

This theorem is referenced by:  pwfilem  9222  incexclem  15796  ramub1lem1  16992  ptcmplem5  24043  onsucsuccmpi  36686