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Theorem eldifpw 7479
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 4547 . . . 4 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 unss1 4152 . . . . 5 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
3 eldifpw.1 . . . . . . 7 𝐶 ∈ V
4 unexg 7461 . . . . . . 7 ((𝐴 ∈ 𝒫 𝐵𝐶 ∈ V) → (𝐴𝐶) ∈ V)
53, 4mpan2 687 . . . . . 6 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
6 elpwg 4541 . . . . . 6 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
75, 6syl 17 . . . . 5 (𝐴 ∈ 𝒫 𝐵 → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ⊆ (𝐵𝐶)))
82, 7syl5ibr 247 . . . 4 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶)))
91, 8mpd 15 . . 3 (𝐴 ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ 𝒫 (𝐵𝐶))
10 elpwi 4547 . . . . 5 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ⊆ 𝐵)
1110unssbd 4161 . . . 4 ((𝐴𝐶) ∈ 𝒫 𝐵𝐶𝐵)
1211con3i 157 . . 3 𝐶𝐵 → ¬ (𝐴𝐶) ∈ 𝒫 𝐵)
139, 12anim12i 612 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
14 eldif 3943 . 2 ((𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴𝐶) ∈ 𝒫 (𝐵𝐶) ∧ ¬ (𝐴𝐶) ∈ 𝒫 𝐵))
1513, 14sylibr 235 1 ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492  cdif 3930  cun 3931  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831
This theorem is referenced by: (None)
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