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Theorem unisuc 4405
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 3287 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4054 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 4335 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 3778 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 3787 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 3784 . . . . 5  |-  U. { A }  =  A
87uneq2i 3268 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2280 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2263 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 270 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   _Vcvv 2740    u. cun 3092    C_ wss 3094   {csn 3581   U.cuni 3768   Tr wtr 4053   suc csuc 4331
This theorem is referenced by:  onunisuci  4443  ordunisuc  4560
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-v 2742  df-un 3099  df-in 3101  df-ss 3108  df-sn 3587  df-pr 3588  df-uni 3769  df-tr 4054  df-suc 4335
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