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Theorem trsspwALT3 44818
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5272. 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 44817, which is the virtual deduction proof trsspwALT 44816 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 5276 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3482 . . . 4 𝑥 ∈ V
32elpw 4609 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3imbitrrdi 252 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 4001 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  𝒫 cpw 4605  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913  df-tr 5266
This theorem is referenced by: (None)
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