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Theorem unisucg 4239
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 Jim Kingdon, 18-Aug-2019.)
Assertion
Ref Expression
unisucg  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )

Proof of Theorem unisucg
StepHypRef Expression
1 df-suc 4196 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
21unieqi 3661 . . . . 5  |-  U. suc  A  =  U. ( A  u.  { A }
)
3 uniun 3670 . . . . 5  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
42, 3eqtri 2108 . . . 4  |-  U. suc  A  =  ( U. A  u.  U. { A }
)
5 unisng 3668 . . . . 5  |-  ( A  e.  V  ->  U. { A }  =  A
)
65uneq2d 3154 . . . 4  |-  ( A  e.  V  ->  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
) )
74, 6syl5eq 2132 . . 3  |-  ( A  e.  V  ->  U. suc  A  =  ( U. A  u.  A ) )
87eqeq1d 2096 . 2  |-  ( A  e.  V  ->  ( U. suc  A  =  A  <-> 
( U. A  u.  A )  =  A ) )
9 df-tr 3935 . . 3  |-  ( Tr  A  <->  U. A  C_  A
)
10 ssequn1 3170 . . 3  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
119, 10bitri 182 . 2  |-  ( Tr  A  <->  ( U. A  u.  A )  =  A )
128, 11syl6rbbr 197 1  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438    u. cun 2997    C_ wss 2999   {csn 3444   U.cuni 3651   Tr wtr 3934   suc csuc 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-suc 4196
This theorem is referenced by:  onsucuni2  4378  nlimsucg  4380  nnsf  11778  peano4nninf  11779
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