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Theorem ordunisssuc 4497
Description: A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordunisssuc  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( U. A  C_  B  <->  A  C_  suc  B ) )

Proof of Theorem ordunisssuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel2 3177 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
2 ordsssuc 4481 . . . . 5  |-  ( ( x  e.  On  /\  Ord  B )  ->  (
x  C_  B  <->  x  e.  suc  B ) )
31, 2sylan 457 . . . 4  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  Ord  B )  ->  ( x  C_  B 
<->  x  e.  suc  B
) )
43an32s 779 . . 3  |-  ( ( ( A  C_  On  /\ 
Ord  B )  /\  x  e.  A )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
54ralbidva 2561 . 2  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( A. x  e.  A  x  C_  B  <->  A. x  e.  A  x  e.  suc  B ) )
6 unissb 3859 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
7 dfss3 3172 . 2  |-  ( A 
C_  suc  B  <->  A. x  e.  A  x  e.  suc  B )
85, 6, 73bitr4g 279 1  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( U. A  C_  B  <->  A  C_  suc  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1686   A.wral 2545    C_ wss 3154   U.cuni 3829   Ord word 4393   Oncon0 4394   suc csuc 4396
This theorem is referenced by:  ordsucuniel  4617  onsucuni  4621  isfinite2  7117  rankbnd2  7543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400
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