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Theorem pwtr 5457
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 5455 . . 3 𝒫 𝐴 = 𝐴
21sseq1i 4012 . 2 ( 𝒫 𝐴 ⊆ 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴)
3 df-tr 5260 . 2 (Tr 𝒫 𝐴 𝒫 𝐴 ⊆ 𝒫 𝐴)
4 dftr4 5266 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
52, 3, 43bitr4ri 304 1 (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wss 3951  𝒫 cpw 4600   cuni 4907  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-tr 5260
This theorem is referenced by:  r1tr  9816  itunitc1  10460
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