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Theorem unisuc 4467
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 3346 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4115 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4397 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3838 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3847 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3844 . . . . 5  |-  U. { A }  =  A
87uneq2i 3327 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2308 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2291 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 270 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1628    e. wcel 1688   _Vcvv 2789    u. cun 3151    C_ wss 3153   {csn 3641   U.cuni 3828   Tr wtr 4114   suc csuc 4393
This theorem is referenced by:  onunisuci  4505  ordunisuc  4622
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rex 2550  df-v 2791  df-un 3158  df-in 3160  df-ss 3167  df-sn 3647  df-pr 3648  df-uni 3829  df-tr 4115  df-suc 4397
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