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Theorem ordunisuc2r 4321
Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
Assertion
Ref Expression
ordunisuc2r  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  A  =  U. A ) )
Distinct variable group:    x, A

Proof of Theorem ordunisuc2r
StepHypRef Expression
1 vex 2622 . . . . . . . . 9  |-  x  e. 
_V
21sucid 4235 . . . . . . . 8  |-  x  e. 
suc  x
3 elunii 3653 . . . . . . . 8  |-  ( ( x  e.  suc  x  /\  suc  x  e.  A
)  ->  x  e.  U. A )
42, 3mpan 415 . . . . . . 7  |-  ( suc  x  e.  A  ->  x  e.  U. A )
54imim2i 12 . . . . . 6  |-  ( ( x  e.  A  ->  suc  x  e.  A )  ->  ( x  e.  A  ->  x  e.  U. A ) )
65alimi 1389 . . . . 5  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x
( x  e.  A  ->  x  e.  U. A
) )
7 df-ral 2364 . . . . 5  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
8 dfss2 3012 . . . . 5  |-  ( A 
C_  U. A  <->  A. x
( x  e.  A  ->  x  e.  U. A
) )
96, 7, 83imtr4i 199 . . . 4  |-  ( A. x  e.  A  suc  x  e.  A  ->  A 
C_  U. A )
109a1i 9 . . 3  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  A 
C_  U. A ) )
11 orduniss 4243 . . 3  |-  ( Ord 
A  ->  U. A  C_  A )
1210, 11jctird 310 . 2  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  ( A  C_  U. A  /\  U. A  C_  A )
) )
13 eqss 3038 . 2  |-  ( A  =  U. A  <->  ( A  C_ 
U. A  /\  U. A  C_  A ) )
1412, 13syl6ibr 160 1  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  A  =  U. A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2997   U.cuni 3648   Ord word 4180   suc csuc 4183
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-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by: (None)
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