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Theorem unisucg 4413
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-tr 4101 . . 3  |-  ( Tr  A  <->  U. A  C_  A
)
2 ssequn1 3305 . . 3  |-  ( U. A  C_  A  <->  ( U. A  u.  A )  =  A )
31, 2bitri 184 . 2  |-  ( Tr  A  <->  ( U. A  u.  A )  =  A )
4 df-suc 4370 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54unieqi 3819 . . . . 5  |-  U. suc  A  =  U. ( A  u.  { A }
)
6 uniun 3828 . . . . 5  |-  U. ( A  u.  { A } )  =  ( U. A  u.  U. { A } )
75, 6eqtri 2198 . . . 4  |-  U. suc  A  =  ( U. A  u.  U. { A }
)
8 unisng 3826 . . . . 5  |-  ( A  e.  V  ->  U. { A }  =  A
)
98uneq2d 3289 . . . 4  |-  ( A  e.  V  ->  ( U. A  u.  U. { A } )  =  ( U. A  u.  A
) )
107, 9eqtrid 2222 . . 3  |-  ( A  e.  V  ->  U. suc  A  =  ( U. A  u.  A ) )
1110eqeq1d 2186 . 2  |-  ( A  e.  V  ->  ( U. suc  A  =  A  <-> 
( U. A  u.  A )  =  A ) )
123, 11bitr4id 199 1  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148    u. cun 3127    C_ wss 3129   {csn 3592   U.cuni 3809   Tr wtr 4100   suc csuc 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4101  df-suc 4370
This theorem is referenced by:  onsucuni2  4562  nlimsucg  4564  ctmlemr  7104  nnnninfeq2  7124  nnsf  14614  peano4nninf  14615
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