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

Theorem onsseleq 4924
Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsseleq  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )

Proof of Theorem onsseleq
StepHypRef Expression
1 eloni 4893 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4893 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordsseleq 4912 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
41, 2, 3syl2an 477 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   Ord word 4882   Oncon0 4883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887
This theorem is referenced by:  onsseli  4997  on0eqel  5000  onmindif2  6641  omword  7229  oeword  7249  oewordi  7250  dffi3  7901  cantnflem1d  8117  cantnflem1  8118  cantnflem1dOLD  8140  cantnflem1OLD  8141  r1ord3g  8207  alephdom  8472  cardaleph  8480  cfsmolem  8660  ttukeylem5  8903  alephreg  8967  inar1  9163  gruina  9206  om2uzlt2i  12040
  Copyright terms: Public domain W3C validator