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Theorem trsspwALT3 44840
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5266. 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 44839, which is the virtual deduction proof trsspwALT 44838 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 5270 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3484 . . . 4 𝑥 ∈ V
32elpw 4604 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3imbitrrdi 252 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 3989 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  𝒫 cpw 4600  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908  df-tr 5260
This theorem is referenced by: (None)
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