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Theorem unisuc 5514
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 3636 . 2  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
2 df-tr 4516 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 df-suc 5444 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
43unieqi 4225 . . . 4  |-  U. suc  A  =  U. ( A  u.  { A }
)
5 uniun 4235 . . . 4  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
6 unisuc.1 . . . . . 6  |-  A  e. 
_V
76unisn 4231 . . . . 5  |-  U. { A }  =  A
87uneq2i 3617 . . . 4  |-  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
)
94, 5, 83eqtri 2455 . . 3  |-  U. suc  A  =  ( U. A  u.  A )
109eqeq1i 2429 . 2  |-  ( U. suc  A  =  A  <->  ( U. A  u.  A )  =  A )
111, 2, 103bitr4i 280 1  |-  ( Tr  A  <->  U. suc  A  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1868   _Vcvv 3081    u. cun 3434    C_ wss 3436   {csn 3996   U.cuni 4216   Tr wtr 4515   suc csuc 5440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-v 3083  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-pr 3999  df-uni 4217  df-tr 4516  df-suc 5444
This theorem is referenced by:  onunisuci  5551  ordunisuc  6669
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