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Theorem unisuc 4409
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 3305 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4099 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4367 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3817 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3826 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3823 . . . . 5  |-  U. { A }  =  A
87uneq2i 3286 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2202 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2185 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 212 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127    C_ wss 3129   {csn 3591   U.cuni 3807   Tr wtr 4098   suc csuc 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-suc 4367
This theorem is referenced by:  onunisuci  4428  ordsucunielexmid  4526  tfrexlem  6328  nnsucuniel  6489
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