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Theorem pwtr 5342
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 5339 . . 3 𝒫 𝐴 = 𝐴
21sseq1i 3999 . 2 ( 𝒫 𝐴 ⊆ 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴)
3 df-tr 5170 . 2 (Tr 𝒫 𝐴 𝒫 𝐴 ⊆ 𝒫 𝐴)
4 dftr4 5174 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
52, 3, 43bitr4ri 305 1 (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wss 3940  𝒫 cpw 4542   cuni 4837  Tr wtr 5169
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-pw 4544  df-sn 4565  df-pr 4567  df-uni 4838  df-tr 5170
This theorem is referenced by:  r1tr  9199  itunitc1  9836
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