Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trsspwALT3 Structured version   Visualization version   GIF version

Theorem trsspwALT3 41513
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5144. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 41512, which is the virtual deduction proof trsspwALT 41511 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 trss 5148 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3447 . . . 4 𝑥 ∈ V
32elpw 4504 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3syl6ibr 255 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 3924 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  wss 3884  𝒫 cpw 4500  Tr wtr 5139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804  df-tr 5140
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator