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Theorem unisuc 4339
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 3247 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4031 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4297 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3750 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3759 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3756 . . . . 5  |-  U. { A }  =  A
87uneq2i 3228 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2165 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2148 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 211 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332    e. wcel 1481   _Vcvv 2687    u. cun 3070    C_ wss 3072   {csn 3528   U.cuni 3740   Tr wtr 4030   suc csuc 4291
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-sn 3534  df-pr 3535  df-uni 3741  df-tr 4031  df-suc 4297
This theorem is referenced by:  onunisuci  4358  ordsucunielexmid  4450  tfrexlem  6235  nnsucuniel  6395
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