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Theorem unipwrVD 39584
Description: Virtual deduction proof of unipwr 39585. (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 3343 . . . . 5 𝑥 ∈ V
21snid 4353 . . . 4 𝑥 ∈ {𝑥}
3 idn1 39310 . . . . 5 (   𝑥𝐴   ▶   𝑥𝐴   )
4 snelpwi 5061 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
53, 4e1a 39372 . . . 4 (   𝑥𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6 elunii 4593 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
72, 5, 6e01an 39437 . . 3 (   𝑥𝐴   ▶   𝑥 𝒫 𝐴   )
87in1 39307 . 2 (𝑥𝐴𝑥 𝒫 𝐴)
98ssriv 3748 1 𝐴 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-vd1 39306
This theorem is referenced by: (None)
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