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Theorem trsspwALT3 44157
Description: Short predicate calculus proof of the left-to-right implication of dftr4 5265. 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 44156, which is the virtual deduction proof trsspwALT 44155 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 5269 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
2 vex 3472 . . . 4 𝑥 ∈ V
32elpw 4601 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 3imbitrrdi 251 . 2 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
54ssrdv 3983 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3943  𝒫 cpw 4597  Tr wtr 5258
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 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903  df-tr 5259
This theorem is referenced by: (None)
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