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Theorem elpwun 7771
Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
Hypothesis
Ref Expression
eldifpw.1 𝐶 ∈ V
Assertion
Ref Expression
elpwun (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)

Proof of Theorem elpwun
StepHypRef Expression
1 elex 3490 . 2 (𝐴 ∈ 𝒫 (𝐵𝐶) → 𝐴 ∈ V)
2 elex 3490 . . 3 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
3 eldifpw.1 . . . 4 𝐶 ∈ V
4 difex2 7762 . . . 4 (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V))
53, 4ax-mp 5 . . 3 (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V)
62, 5sylibr 233 . 2 ((𝐴𝐶) ∈ 𝒫 𝐵𝐴 ∈ V)
7 elpwg 4606 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ 𝐴 ⊆ (𝐵𝐶)))
8 uncom 4152 . . . . . 6 (𝐵𝐶) = (𝐶𝐵)
98sseq2i 4009 . . . . 5 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
10 ssundif 4488 . . . . 5 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
119, 10bitri 275 . . . 4 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ⊆ 𝐵)
12 difexg 5329 . . . . 5 (𝐴 ∈ V → (𝐴𝐶) ∈ V)
13 elpwg 4606 . . . . 5 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
1412, 13syl 17 . . . 4 (𝐴 ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
1511, 14bitr4id 290 . . 3 (𝐴 ∈ V → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
167, 15bitrd 279 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
171, 6, 16pm5.21nii 378 1 (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  Vcvv 3471  cdif 3944  cun 3945  wss 3947  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by:  pwfilem  9202  pwfilemOLD  9371  elrfi  42114  dssmapnvod  43450
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