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Theorem unipwrVD 44169
Description: Virtual deduction proof of unipwr 44170. (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 3472 . . . . 5 𝑥 ∈ V
21snid 4659 . . . 4 𝑥 ∈ {𝑥}
3 idn1 43911 . . . . 5 (   𝑥𝐴   ▶   𝑥𝐴   )
4 snelpwi 5436 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
53, 4e1a 43964 . . . 4 (   𝑥𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6 elunii 4907 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
72, 5, 6e01an 44029 . . 3 (   𝑥𝐴   ▶   𝑥 𝒫 𝐴   )
87in1 43908 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
98ssriv 3981 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wss 3943  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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903  df-vd1 43907
This theorem is referenced by: (None)
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