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Theorem pwtr 4174
 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 4172 . . 3 𝒫 𝐴 = 𝐴
21sseq1i 3150 . 2 ( 𝒫 𝐴 ⊆ 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴)
3 df-tr 4059 . 2 (Tr 𝒫 𝐴 𝒫 𝐴 ⊆ 𝒫 𝐴)
4 dftr4 4063 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
52, 3, 43bitr4ri 212 1 (Tr 𝐴 ↔ Tr 𝒫 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ⊆ wss 3098  𝒫 cpw 3539  ∪ cuni 3768  Tr wtr 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-uni 3769  df-tr 4059 This theorem is referenced by: (None)
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