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Theorem unipwr 42453
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5366. The proof of this theorem was automatically generated from unipwrVD 42452 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
unipwr 𝐴 𝒫 𝐴

Proof of Theorem unipwr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3436 . . . 4 𝑥 ∈ V
21snid 4597 . . 3 𝑥 ∈ {𝑥}
3 snelpwi 5360 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 elunii 4844 . . 3 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
52, 3, 4sylancr 587 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
65ssriv 3925 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3887  𝒫 cpw 4533  {csn 4561   cuni 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840
This theorem is referenced by: (None)
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