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Theorem ordunisuc2r 4509
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 2740 . . . . . . . . 9  |-  x  e. 
_V
21sucid 4413 . . . . . . . 8  |-  x  e. 
suc  x
3 elunii 3812 . . . . . . . 8  |-  ( ( x  e.  suc  x  /\  suc  x  e.  A
)  ->  x  e.  U. A )
42, 3mpan 424 . . . . . . 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 1455 . . . . 5  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x
( x  e.  A  ->  x  e.  U. A
) )
7 df-ral 2460 . . . . 5  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
8 dfss2 3144 . . . . 5  |-  ( A 
C_  U. A  <->  A. x
( x  e.  A  ->  x  e.  U. A
) )
96, 7, 83imtr4i 201 . . . 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 4421 . . 3  |-  ( Ord 
A  ->  U. A  C_  A )
1210, 11jctird 317 . 2  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  ( A  C_  U. A  /\  U. A  C_  A )
) )
13 eqss 3170 . 2  |-  ( A  =  U. A  <->  ( A  C_ 
U. A  /\  U. A  C_  A ) )
1412, 13syl6ibr 162 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 104   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   U.cuni 3807   Ord word 4358   suc csuc 4361
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-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-uni 3808  df-tr 4099  df-iord 4362  df-suc 4367
This theorem is referenced by: (None)
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