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Theorem unisuc 4617
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 3477 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4263 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4547 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3985 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3994 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3991 . . . . 5  |-  U. { A }  =  A
87uneq2i 3458 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2428 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2411 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 269 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   {csn 3774   U.cuni 3975   Tr wtr 4262   suc csuc 4543
This theorem is referenced by:  onunisuci  4654  ordunisuc  4771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-sn 3780  df-pr 3781  df-uni 3976  df-tr 4263  df-suc 4547
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