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Theorem eldifpw 7018
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 4201 . . . 4 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 unss1 3815 . . . . 5 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 ∈ V
4 unexg 7001 . . . . . . 7 ((𝐴 ∈ 𝒫 𝐵𝐶 ∈ V) → (𝐴𝐶) ∈ V)
53, 4mpan2 707 . . . . . 6 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
6 elpwg 4199 . . . . . 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 4201 . . . . 5 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ⊆ 𝐵)
1110unssbd 3824 . . . 4 ((𝐴𝐶) ∈ 𝒫 𝐵𝐶𝐵)
1211con3i 150 . . 3 𝐶𝐵 → ¬ (𝐴𝐶) ∈ 𝒫 𝐵)
139, 12anim12i 589 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
14 eldif 3617 . 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 2030  Vcvv 3231  cdif 3604  cun 3605  wss 3607  𝒫 cpw 4191
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
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-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469
This theorem is referenced by: (None)
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