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Theorem unipwr 41029
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5338. The proof of this theorem was automatically generated from unipwrVD 41028 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 3502 . . . 4 𝑥 ∈ V
21snid 4597 . . 3 𝑥 ∈ {𝑥}
3 snelpwi 5332 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 elunii 4841 . . 3 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
52, 3, 4sylancr 587 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
65ssriv 3974 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3939  𝒫 cpw 4541  {csn 4563   cuni 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543  df-sn 4564  df-pr 4566  df-uni 4837
This theorem is referenced by: (None)
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