Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unipwrVD Structured version   Visualization version   GIF version

Theorem unipwrVD 41031
 Description: Virtual deduction proof of unipwr 41032. (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 3503 . . . . 5 𝑥 ∈ V
21snid 4598 . . . 4 𝑥 ∈ {𝑥}
3 idn1 40773 . . . . 5 (   𝑥𝐴   ▶   𝑥𝐴   )
4 snelpwi 5333 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
53, 4e1a 40826 . . . 4 (   𝑥𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6 elunii 4842 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
72, 5, 6e01an 40891 . . 3 (   𝑥𝐴   ▶   𝑥 𝒫 𝐴   )
87in1 40770 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
98ssriv 3975 1 𝐴 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2107   ⊆ wss 3940  𝒫 cpw 4542  {csn 4564  ∪ cuni 4837 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-pw 4544  df-sn 4565  df-pr 4567  df-uni 4838  df-vd1 40769 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator