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Theorem unisuc 4484
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 3358 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4130 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4414 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3853 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3862 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3859 . . . . 5  |-  U. { A }  =  A
87uneq2i 3339 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2320 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2303 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 268 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653   U.cuni 3843   Tr wtr 4129   suc csuc 4410
This theorem is referenced by:  onunisuci  4522  ordunisuc  4639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-pr 3660  df-uni 3844  df-tr 4130  df-suc 4414
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