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Theorem unipwrVD 39817
Description: Virtual deduction proof of unipwr 39818. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
unipwrVD 𝐴 𝒫 𝐴

Proof of Theorem unipwrVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3387 . . . . 5 𝑥 ∈ V
21snid 4399 . . . 4 𝑥 ∈ {𝑥}
3 idn1 39549 . . . . 5 (   𝑥𝐴   ▶   𝑥𝐴   )
4 snelpwi 5102 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
53, 4e1a 39611 . . . 4 (   𝑥𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6 elunii 4632 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
72, 5, 6e01an 39676 . . 3 (   𝑥𝐴   ▶   𝑥 𝒫 𝐴   )
87in1 39546 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
98ssriv 3801 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  wss 3768  𝒫 cpw 4348  {csn 4367   cuni 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-pw 4350  df-sn 4368  df-pr 4370  df-uni 4628  df-vd1 39545
This theorem is referenced by: (None)
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