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Theorem ordunisssuc 4932
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 3462 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
2 ordsssuc 4916 . . . . 5  |-  ( ( x  e.  On  /\  Ord  B )  ->  (
x  C_  B  <->  x  e.  suc  B ) )
31, 2sylan 471 . . . 4  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  Ord  B )  ->  ( x  C_  B 
<->  x  e.  suc  B
) )
43an32s 802 . . 3  |-  ( ( ( A  C_  On  /\ 
Ord  B )  /\  x  e.  A )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
54ralbidva 2844 . 2  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( A. x  e.  A  x  C_  B  <->  A. x  e.  A  x  e.  suc  B ) )
6 unissb 4234 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
7 dfss3 3457 . 2  |-  ( A 
C_  suc  B  <->  A. x  e.  A  x  e.  suc  B )
85, 6, 73bitr4g 288 1  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( U. A  C_  B  <->  A  C_  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   A.wral 2799    C_ wss 3439   U.cuni 4202   Ord word 4829   Oncon0 4830   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836
This theorem is referenced by:  ordsucuniel  6548  onsucuni  6552  isfinite2  7684  rankbnd2  8190
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