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Theorem trsspwALT3 44324
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5267. 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 44323, which is the virtual deduction proof trsspwALT 44322 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 5271 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3467 . . . 4 𝑥 ∈ V
32elpw 4602 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3imbitrrdi 251 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 3978 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3939  𝒫 cpw 4598  Tr wtr 5260
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-v 3465  df-ss 3956  df-pw 4600  df-uni 4904  df-tr 5261
This theorem is referenced by: (None)
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