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Theorem unipwr 39583
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5046. The proof of this theorem was automatically generated from unipwrVD 39582 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 3354 . . . 4 𝑥 ∈ V
21snid 4347 . . 3 𝑥 ∈ {𝑥}
3 snelpwi 5040 . . 3 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 elunii 4579 . . 3 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
52, 3, 4sylancr 575 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
65ssriv 3756 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  wss 3723  𝒫 cpw 4297  {csn 4316   cuni 4574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-uni 4575
This theorem is referenced by: (None)
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