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Theorem ordunisuc2r 4425
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 2684 . . . . . . . . 9  |-  x  e. 
_V
21sucid 4334 . . . . . . . 8  |-  x  e. 
suc  x
3 elunii 3736 . . . . . . . 8  |-  ( ( x  e.  suc  x  /\  suc  x  e.  A
)  ->  x  e.  U. A )
42, 3mpan 420 . . . . . . 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 1431 . . . . 5  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x
( x  e.  A  ->  x  e.  U. A
) )
7 df-ral 2419 . . . . 5  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
8 dfss2 3081 . . . . 5  |-  ( A 
C_  U. A  <->  A. x
( x  e.  A  ->  x  e.  U. A
) )
96, 7, 83imtr4i 200 . . . 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 4342 . . 3  |-  ( Ord 
A  ->  U. A  C_  A )
1210, 11jctird 315 . 2  |-  ( Ord 
A  ->  ( A. x  e.  A  suc  x  e.  A  ->  ( A  C_  U. A  /\  U. A  C_  A )
) )
13 eqss 3107 . 2  |-  ( A  =  U. A  <->  ( A  C_ 
U. A  /\  U. A  C_  A ) )
1412, 13syl6ibr 161 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 103   A.wal 1329    = wceq 1331    e. wcel 1480   A.wral 2414    C_ wss 3066   U.cuni 3731   Ord word 4279   suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283  df-suc 4288
This theorem is referenced by: (None)
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