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Theorem unisuc 4644
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 3504 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4290 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4574 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 4012 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 4021 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 4018 . . . . 5  |-  U. { A }  =  A
87uneq2i 3485 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2454 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2437 . 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 1652    e. wcel 1725   _Vcvv 2943    u. cun 3305    C_ wss 3307   {csn 3801   U.cuni 4002   Tr wtr 4289   suc csuc 4570
This theorem is referenced by:  onunisuci  4681  ordunisuc  4798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-rex 2698  df-v 2945  df-un 3312  df-in 3314  df-ss 3321  df-sn 3807  df-pr 3808  df-uni 4003  df-tr 4290  df-suc 4574
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