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Theorem ordunisssuc 4625
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 3287 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
2 ordsssuc 4609 . . . . 5  |-  ( ( x  e.  On  /\  Ord  B )  ->  (
x  C_  B  <->  x  e.  suc  B ) )
31, 2sylan 458 . . . 4  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  Ord  B )  ->  ( x  C_  B 
<->  x  e.  suc  B
) )
43an32s 780 . . 3  |-  ( ( ( A  C_  On  /\ 
Ord  B )  /\  x  e.  A )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
54ralbidva 2666 . 2  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( A. x  e.  A  x  C_  B  <->  A. x  e.  A  x  e.  suc  B ) )
6 unissb 3988 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
7 dfss3 3282 . 2  |-  ( A 
C_  suc  B  <->  A. x  e.  A  x  e.  suc  B )
85, 6, 73bitr4g 280 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 177    /\ wa 359    e. wcel 1717   A.wral 2650    C_ wss 3264   U.cuni 3958   Ord word 4522   Oncon0 4523   suc csuc 4525
This theorem is referenced by:  ordsucuniel  4745  onsucuni  4749  isfinite2  7302  rankbnd2  7729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529
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