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Theorem pwtr 3946
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 3944 . . 3 𝒫 A = A
21sseq1i 2963 . 2 ( 𝒫 A ⊆ 𝒫 AA ⊆ 𝒫 A)
3 df-tr 3846 . 2 (Tr 𝒫 A 𝒫 A ⊆ 𝒫 A)
4 dftr4 3850 . 2 (Tr AA ⊆ 𝒫 A)
52, 3, 43bitr4ri 202 1 (Tr A ↔ Tr 𝒫 A)
Colors of variables: wff set class
Syntax hints:  wb 98  wss 2911  𝒫 cpw 3351   cuni 3571  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846
This theorem is referenced by: (None)
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