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Theorem unisucg 4294
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 4251 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
21unieqi 3710 . . . . 5  |-  U. suc  A  =  U. ( A  u.  { A }
)
3 uniun 3719 . . . . 5  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
42, 3eqtri 2133 . . . 4  |-  U. suc  A  =  ( U. A  u.  U. { A }
)
5 unisng 3717 . . . . 5  |-  ( A  e.  V  ->  U. { A }  =  A
)
65uneq2d 3194 . . . 4  |-  ( A  e.  V  ->  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
) )
74, 6syl5eq 2157 . . 3  |-  ( A  e.  V  ->  U. suc  A  =  ( U. A  u.  A ) )
87eqeq1d 2121 . 2  |-  ( A  e.  V  ->  ( U. suc  A  =  A  <-> 
( U. A  u.  A )  =  A ) )
9 df-tr 3985 . . 3  |-  ( Tr  A  <->  U. A  C_  A
)
10 ssequn1 3210 . . 3  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
119, 10bitri 183 . 2  |-  ( Tr  A  <->  ( U. A  u.  A )  =  A )
128, 11syl6rbbr 198 1  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1312    e. wcel 1461    u. cun 3033    C_ wss 3035   {csn 3491   U.cuni 3700   Tr wtr 3984   suc csuc 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-uni 3701  df-tr 3985  df-suc 4251
This theorem is referenced by:  onsucuni2  4437  nlimsucg  4439  ctmlemr  6943  nnsf  12880  peano4nninf  12881
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