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Theorem elpwunsn 4368
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 3725 . 2 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2 elpwg 4310 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
3 dfss3 3733 . . . . . . 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 502 . . . 4 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥𝐴 𝑥𝐵)
7 rexnal 3133 . . . 4 (∃𝑥𝐴 ¬ 𝑥𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵)
86, 7sylibr 224 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
9 elpwi 4312 . . . . . . . . . 10 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10 ssel 3738 . . . . . . . . . 10 (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})))
11 elun 3896 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥𝐵𝑥 ∈ {𝐶}))
12 elsni 4338 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
1312orim2i 541 . . . . . . . . . . . . . 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 2827 . . . . . . . . . 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 3168 . . . 4 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴))
2625imp 444 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥𝐴 ¬ 𝑥𝐵) → 𝐶𝐴)
278, 26syldan 488 . 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 1632  wcel 2139  wral 3050  wrex 3051  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pw 4304  df-sn 4322
This theorem is referenced by:  pwfilem  8425  incexclem  14767  ramub1lem1  15932  ptcmplem5  22061  onsucsuccmpi  32748
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