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Theorem elpwun 7475
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 3462 . 2 (𝐴 ∈ 𝒫 (𝐵𝐶) → 𝐴 ∈ V)
2 elex 3462 . . 3 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
3 eldifpw.1 . . . 4 𝐶 ∈ V
4 difex2 7466 . . . 4 (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V))
53, 4ax-mp 5 . . 3 (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V)
62, 5sylibr 237 . 2 ((𝐴𝐶) ∈ 𝒫 𝐵𝐴 ∈ V)
7 elpwg 4503 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ 𝐴 ⊆ (𝐵𝐶)))
8 difexg 5198 . . . . 5 (𝐴 ∈ V → (𝐴𝐶) ∈ V)
9 elpwg 4503 . . . . 5 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
108, 9syl 17 . . . 4 (𝐴 ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
11 uncom 4083 . . . . . 6 (𝐵𝐶) = (𝐶𝐵)
1211sseq2i 3947 . . . . 5 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
13 ssundif 4394 . . . . 5 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
1412, 13bitri 278 . . . 4 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ⊆ 𝐵)
1510, 14syl6rbbr 293 . . 3 (𝐴 ∈ V → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
167, 15bitrd 282 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
171, 6, 16pm5.21nii 383 1 (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2112  Vcvv 3444  cdif 3881  cun 3882  wss 3884  𝒫 cpw 4500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4804
This theorem is referenced by:  pwfilem  8806  elrfi  39622  dssmapnvod  40708
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