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Theorem unipwr 38551
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4879. The proof of this theorem was automatically generated from unipwrVD 38550 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 3189 . . . 4 𝑥 ∈ V
21snid 4179 . . 3 𝑥 ∈ {𝑥}
3 snelpwi 4873 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 elunii 4407 . . 3 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
52, 3, 4sylancr 694 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
65ssriv 3587 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151  df-uni 4403
This theorem is referenced by: (None)
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