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Theorem unisuc 4468
Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisuc.1  |-  A  e. 
_V
Assertion
Ref Expression
unisuc  |-  ( Tr  A  <->  U. suc  A  =  A )

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3345 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4114 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4398 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3837 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3846 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3843 . . . . 5  |-  U. { A }  =  A
87uneq2i 3326 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2307 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2290 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 268 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640   U.cuni 3827   Tr wtr 4113   suc csuc 4394
This theorem is referenced by:  onunisuci  4506  ordunisuc  4623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-pr 3647  df-uni 3828  df-tr 4114  df-suc 4398
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