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

Theorem snelpwrVD 42433
Description: Virtual deduction proof of snelpwi 5364. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snelpwrVD (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwrVD
StepHypRef Expression
1 snex 5358 . . 3 {𝐴} ∈ V
2 idn1 42176 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
3 snssi 4747 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
42, 3e1a 42229 . . 3 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
5 elpwg 4542 . . . 4 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
65biimprd 247 . . 3 ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
71, 4, 6e01 42293 . 2 (   𝐴𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
87in1 42173 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3431  wss 3892  𝒫 cpw 4539  {csn 4567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-pw 4541  df-sn 4568  df-pr 4570  df-vd1 42172
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator