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Theorem elpwunsn 4170
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 3549 . 2 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2 elpwg 4115 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
3 dfss3 3557 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
42, 3syl6bb 274 . . . . . 6 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵))
54notbid 306 . . . . 5 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵))
65biimpa 499 . . . 4 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥𝐴 𝑥𝐵)
7 rexnal 2977 . . . 4 (∃𝑥𝐴 ¬ 𝑥𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵)
86, 7sylibr 222 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
9 elpwi 4116 . . . . . . . . . 10 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10 ssel 3561 . . . . . . . . . 10 (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})))
11 elun 3714 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥𝐵𝑥 ∈ {𝐶}))
12 elsni 4141 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
1312orim2i 538 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑥 ∈ {𝐶}) → (𝑥𝐵𝑥 = 𝐶))
1413ord 390 . . . . . . . . . . . . 13 ((𝑥𝐵𝑥 ∈ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1511, 14sylbi 205 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∪ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1615imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → (𝑥𝐴 → (¬ 𝑥𝐵𝑥 = 𝐶)))
1716impd 445 . . . . . . . . . 10 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
189, 10, 173syl 18 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
19 eleq1 2675 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2019biimpd 217 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2118, 20syl6 34 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴𝐶𝐴)))
2221expd 450 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥𝐴𝐶𝐴))))
2322com4r 91 . . . . . 6 (𝑥𝐴 → (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵𝐶𝐴))))
2423pm2.43b 52 . . . . 5 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝑥𝐴 → (¬ 𝑥𝐵𝐶𝐴)))
2524rexlimdv 3011 . . . 4 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴))
2625imp 443 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥𝐴 ¬ 𝑥𝐵) → 𝐶𝐴)
278, 26syldan 485 . 2 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶𝐴)
281, 27sylbi 205 1 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wcel 1976  wral 2895  wrex 2896  cdif 3536  cun 3537  wss 3539  𝒫 cpw 4107  {csn 4124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pw 4109  df-sn 4125
This theorem is referenced by:  pwfilem  8120  incexclem  14355  ramub1lem1  15516  ptcmplem5  21617  onsucsuccmpi  31405
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