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Theorem eldifpw 7141
Description: Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
Hypothesis
Ref Expression
eldifpw.1 𝐶 ∈ V
Assertion
Ref Expression
eldifpw ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))

Proof of Theorem eldifpw
StepHypRef Expression
1 elpwi 4312 . . . 4 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 unss1 3925 . . . . 5 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 ∈ V
4 unexg 7124 . . . . . . 7 ((𝐴 ∈ 𝒫 𝐵𝐶 ∈ V) → (𝐴𝐶) ∈ V)
53, 4mpan2 709 . . . . . 6 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
6 elpwg 4310 . . . . . 6 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
75, 6syl 17 . . . . 5 (𝐴 ∈ 𝒫 𝐵 → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
82, 7syl5ibr 236 . . . 4 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶)))
91, 8mpd 15 . . 3 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶))
10 elpwi 4312 . . . . 5 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ⊆ 𝐵)
1110unssbd 3934 . . . 4 ((𝐴𝐶) ∈ 𝒫 𝐵𝐶𝐵)
1211con3i 150 . . 3 𝐶𝐵 → ¬ (𝐴𝐶) ∈ 𝒫 𝐵)
139, 12anim12i 591 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
14 eldif 3725 . 2 ((𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
1513, 14sylibr 224 1 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2139  Vcvv 3340  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302
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-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
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-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by: (None)
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