MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordunisssuc Structured version   Unicode version

Theorem ordunisssuc 4676
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 3335 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
2 ordsssuc 4660 . . . . 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 2713 . 2  |-  ( ( A  C_  On  /\  Ord  B )  ->  ( A. x  e.  A  x  C_  B  <->  A. x  e.  A  x  e.  suc  B ) )
6 unissb 4037 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
7 dfss3 3330 . 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 1725   A.wral 2697    C_ wss 3312   U.cuni 4007   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  ordsucuniel  4796  onsucuni  4800  isfinite2  7357  rankbnd2  7787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
  Copyright terms: Public domain W3C validator