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Theorem eldifpw 7707
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 4572 . . . 4 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 unss1 4144 . . . . 5 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 ∈ V
4 unexg 7688 . . . . . . 7 ((𝐴 ∈ 𝒫 𝐵𝐶 ∈ V) → (𝐴𝐶) ∈ V)
53, 4mpan2 690 . . . . . 6 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
6 elpwg 4568 . . . . . 6 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
75, 6syl 17 . . . . 5 (𝐴 ∈ 𝒫 𝐵 → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
82, 7syl5ibr 246 . . . 4 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶)))
91, 8mpd 15 . . 3 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶))
10 elpwi 4572 . . . . 5 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ⊆ 𝐵)
1110unssbd 4153 . . . 4 ((𝐴𝐶) ∈ 𝒫 𝐵𝐶𝐵)
1211con3i 154 . . 3 𝐶𝐵 → ¬ (𝐴𝐶) ∈ 𝒫 𝐵)
139, 12anim12i 614 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
14 eldif 3925 . 2 ((𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
1513, 14sylibr 233 1 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107  Vcvv 3448  cdif 3912  cun 3913  wss 3915  𝒫 cpw 4565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4871
This theorem is referenced by: (None)
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