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Theorem unisucg 4177
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 4134 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
21unieqi 3619 . . . . 5  |-  U. suc  A  =  U. ( A  u.  { A }
)
3 uniun 3628 . . . . 5  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
42, 3eqtri 2102 . . . 4  |-  U. suc  A  =  ( U. A  u.  U. { A }
)
5 unisng 3626 . . . . 5  |-  ( A  e.  V  ->  U. { A }  =  A
)
65uneq2d 3127 . . . 4  |-  ( A  e.  V  ->  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
) )
74, 6syl5eq 2126 . . 3  |-  ( A  e.  V  ->  U. suc  A  =  ( U. A  u.  A ) )
87eqeq1d 2090 . 2  |-  ( A  e.  V  ->  ( U. suc  A  =  A  <-> 
( U. A  u.  A )  =  A ) )
9 df-tr 3884 . . 3  |-  ( Tr  A  <->  U. A  C_  A
)
10 ssequn1 3143 . . 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 1285    e. wcel 1434    u. cun 2972    C_ wss 2974   {csn 3406   U.cuni 3609   Tr wtr 3883   suc csuc 4128
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-suc 4134
This theorem is referenced by:  onsucuni2  4315  nlimsucg  4317
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