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Theorem trsspwALT3 38527
Description: Short predicate calculus proof of the left-to-right implication of dftr4 4717. 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 38526, which is the virtual deduction proof trsspwALT 38525 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 4721 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3189 . . . 4 𝑥 ∈ V
32elpw 4136 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3syl6ibr 242 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 3589 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3555  𝒫 cpw 4130  Tr wtr 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-tr 4713
This theorem is referenced by: (None)
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