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Theorem unisuc 4963
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 3670 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4551 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4893 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 4260 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 4270 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 4266 . . . . 5  |-  U. { A }  =  A
87uneq2i 3651 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2490 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2464 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 277 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   {csn 4032   U.cuni 4251   Tr wtr 4550   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-pr 4035  df-uni 4252  df-tr 4551  df-suc 4893
This theorem is referenced by:  onunisuci  5000  ordunisuc  6666
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