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Theorem pwtr 5077
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 5074 . . 3 𝒫 𝐴 = 𝐴
21sseq1i 3789 . 2 ( 𝒫 𝐴 ⊆ 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴)
3 df-tr 4912 . 2 (Tr 𝒫 𝐴 𝒫 𝐴 ⊆ 𝒫 𝐴)
4 dftr4 4916 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
52, 3, 43bitr4ri 295 1 (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wss 3732  𝒫 cpw 4315   cuni 4594  Tr wtr 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-pw 4317  df-sn 4335  df-pr 4337  df-uni 4595  df-tr 4912
This theorem is referenced by:  r1tr  8854  itunitc1  9495
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