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Theorem ordunisssuc 4495
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 3175 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
2 ordsssuc 4479 . . . . 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 2559 . 2  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( A. x  e.  A  x  C_  B  <->  A. x  e.  A  x  e.  suc  B ) )
6 unissb 3857 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
7 dfss3 3170 . 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 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordsucuniel  4615  onsucuni  4619  isfinite2  7115  rankbnd2  7541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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