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Theorem elpwunsn 4256
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) → 𝐶𝐴)

Proof of Theorem elpwunsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 3617 . 2 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2 elpwg 4199 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
3 dfss3 3625 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
42, 3syl6bb 276 . . . . . 6 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵))
54notbid 307 . . . . 5 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵))
65biimpa 500 . . . 4 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥𝐴 𝑥𝐵)
7 rexnal 3024 . . . 4 (∃𝑥𝐴 ¬ 𝑥𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵)
86, 7sylibr 224 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
9 elpwi 4201 . . . . . . . . . 10 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10 ssel 3630 . . . . . . . . . 10 (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})))
11 elun 3786 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥𝐵𝑥 ∈ {𝐶}))
12 elsni 4227 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
1312orim2i 539 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑥 ∈ {𝐶}) → (𝑥𝐵𝑥 = 𝐶))
1413ord 391 . . . . . . . . . . . . 13 ((𝑥𝐵𝑥 ∈ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1511, 14sylbi 207 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∪ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1615imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → (𝑥𝐴 → (¬ 𝑥𝐵𝑥 = 𝐶)))
1716impd 446 . . . . . . . . . 10 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
189, 10, 173syl 18 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
19 eleq1 2718 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2019biimpd 219 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2118, 20syl6 35 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴𝐶𝐴)))
2221expd 451 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥𝐴𝐶𝐴))))
2322com4r 94 . . . . . 6 (𝑥𝐴 → (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵𝐶𝐴))))
2423pm2.43b 55 . . . . 5 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵𝐶𝐴)))
2524rexlimdv 3059 . . . 4 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴))
2625imp 444 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥𝐴 ¬ 𝑥𝐵) → 𝐶𝐴)
278, 26syldan 486 . 2 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶𝐴)
281, 27sylbi 207 1 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cdif 3604  cun 3605  wss 3607  𝒫 cpw 4191  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pw 4193  df-sn 4211
This theorem is referenced by:  pwfilem  8301  incexclem  14612  ramub1lem1  15777  ptcmplem5  21907  onsucsuccmpi  32567
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