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Theorem elpwun 7142
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 3352 . 2 (𝐴 ∈ 𝒫 (𝐵𝐶) → 𝐴 ∈ V)
2 elex 3352 . . 3 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
3 eldifpw.1 . . . 4 𝐶 ∈ V
4 difex2 7134 . . . 4 (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V))
53, 4ax-mp 5 . . 3 (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V)
62, 5sylibr 224 . 2 ((𝐴𝐶) ∈ 𝒫 𝐵𝐴 ∈ V)
7 elpwg 4310 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ 𝐴 ⊆ (𝐵𝐶)))
8 difexg 4960 . . . . 5 (𝐴 ∈ V → (𝐴𝐶) ∈ V)
9 elpwg 4310 . . . . 5 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
108, 9syl 17 . . . 4 (𝐴 ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
11 uncom 3900 . . . . . 6 (𝐵𝐶) = (𝐶𝐵)
1211sseq2i 3771 . . . . 5 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
13 ssundif 4196 . . . . 5 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
1412, 13bitri 264 . . . 4 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ⊆ 𝐵)
1510, 14syl6rbbr 279 . . 3 (𝐴 ∈ V → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
167, 15bitrd 268 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
171, 6, 16pm5.21nii 367 1 (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2139  Vcvv 3340  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by:  pwfilem  8425  elrfi  37759  dssmapnvod  38816
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