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Theorem dftr4 4090
Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr4  |-  ( Tr  A  <->  A  C_  ~P A
)

Proof of Theorem dftr4
StepHypRef Expression
1 df-tr 4086 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3955 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 186 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    C_ wss 3121   ~Pcpw 3564   U.cuni 3794   Tr wtr 4085
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-uni 3795  df-tr 4086
This theorem is referenced by:  tr0  4096  pwtr  4202  pw1on  7190
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