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Theorem elpwun 6925
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 3203 . 2 (𝐴 ∈ 𝒫 (𝐵𝐶) → 𝐴 ∈ V)
2 elex 3203 . . 3 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
3 eldifpw.1 . . . 4 𝐶 ∈ V
4 difex2 6919 . . . 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 4143 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ 𝐴 ⊆ (𝐵𝐶)))
8 difexg 4773 . . . . 5 (𝐴 ∈ V → (𝐴𝐶) ∈ V)
9 elpwg 4143 . . . . 5 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
108, 9syl 17 . . . 4 (𝐴 ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
11 uncom 3740 . . . . . 6 (𝐵𝐶) = (𝐶𝐵)
1211sseq2i 3614 . . . . 5 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
13 ssundif 4029 . . . . 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 368 1 (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1992  Vcvv 3191  cdif 3557  cun 3558  wss 3560  𝒫 cpw 4135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408
This theorem is referenced by:  pwfilem  8205  elrfi  36723  dssmapnvod  37782
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