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Theorem unipwr 43429
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5444. The proof of this theorem was automatically generated from unipwrVD 43428 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 3478 . . . 4 𝑥 ∈ V
21snid 4659 . . 3 𝑥 ∈ {𝑥}
3 snelpwi 5437 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 elunii 4907 . . 3 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
52, 3, 4sylancr 587 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
65ssriv 3983 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3945  𝒫 cpw 4597  {csn 4623   cuni 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5293  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3950  df-in 3952  df-ss 3962  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903
This theorem is referenced by: (None)
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