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Theorem elpwunsn 4575
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 3854 . 2 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) ↔ (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵))
2 elpwg 4492 . . . . . . 7 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
3 dfss3 3866 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
42, 3bitrdi 290 . . . . . 6 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (𝐴 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵))
54notbid 321 . . . . 5 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (¬ 𝐴 ∈ 𝒫 𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵))
65biimpa 480 . . . 4 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ¬ ∀𝑥𝐴 𝑥𝐵)
7 rexnal 3151 . . . 4 (∃𝑥𝐴 ¬ 𝑥𝐵 ↔ ¬ ∀𝑥𝐴 𝑥𝐵)
86, 7sylibr 237 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
9 elpwi 4498 . . . . . . . . . 10 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → 𝐴 ⊆ (𝐵 ∪ {𝐶}))
10 ssel 3871 . . . . . . . . . 10 (𝐴 ⊆ (𝐵 ∪ {𝐶}) → (𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})))
11 elun 4040 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑥𝐵𝑥 ∈ {𝐶}))
12 elsni 4534 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝐶} → 𝑥 = 𝐶)
1312orim2i 910 . . . . . . . . . . . . . 14 ((𝑥𝐵𝑥 ∈ {𝐶}) → (𝑥𝐵𝑥 = 𝐶))
1413ord 863 . . . . . . . . . . . . 13 ((𝑥𝐵𝑥 ∈ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1511, 14sylbi 220 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∪ {𝐶}) → (¬ 𝑥𝐵𝑥 = 𝐶))
1615imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → (𝑥𝐴 → (¬ 𝑥𝐵𝑥 = 𝐶)))
1716impd 414 . . . . . . . . . 10 ((𝑥𝐴𝑥 ∈ (𝐵 ∪ {𝐶})) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
189, 10, 173syl 18 . . . . . . . . 9 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 = 𝐶))
19 eleq1 2820 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2019biimpd 232 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
2118, 20syl6 35 . . . . . . . 8 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥𝐴𝐶𝐴)))
2221expd 419 . . . . . . 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 3193 . . . 4 (𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) → (∃𝑥𝐴 ¬ 𝑥𝐵𝐶𝐴))
2625imp 410 . . 3 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ∃𝑥𝐴 ¬ 𝑥𝐵) → 𝐶𝐴)
278, 26syldan 594 . 2 ((𝐴 ∈ 𝒫 (𝐵 ∪ {𝐶}) ∧ ¬ 𝐴 ∈ 𝒫 𝐵) → 𝐶𝐴)
281, 27sylbi 220 1 (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 846   = wceq 1542  wcel 2113  wral 3053  wrex 3054  cdif 3841  cun 3842  wss 3844  𝒫 cpw 4489  {csn 4517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pw 4491  df-sn 4518
This theorem is referenced by:  pwfilem  8776  pwfilemOLD  8892  incexclem  15285  ramub1lem1  16463  ptcmplem5  22808  onsucsuccmpi  34270
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