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Theorem pwtr 5453
Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
Assertion
Ref Expression
pwtr (Tr 𝐴 ↔ Tr 𝒫 𝐴)

Proof of Theorem pwtr
StepHypRef Expression
1 unipw 5451 . . 3 𝒫 𝐴 = 𝐴
21sseq1i 4011 . 2 ( 𝒫 𝐴 ⊆ 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴)
3 df-tr 5267 . 2 (Tr 𝒫 𝐴 𝒫 𝐴 ⊆ 𝒫 𝐴)
4 dftr4 5273 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
52, 3, 43bitr4ri 304 1 (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wss 3949  𝒫 cpw 4603   cuni 4909  Tr wtr 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-tr 5267
This theorem is referenced by:  r1tr  9771  itunitc1  10415
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