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

Theorem onelss 4583
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
onelss  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4551 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4557 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
32ex 424 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  B  C_  A ) )
41, 3syl 16 1  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    C_ wss 3280   Ord word 4540   Oncon0 4541
This theorem is referenced by:  ordunidif  4589  onelssi  4649  ssorduni  4725  suceloni  4752  tfisi  4797  tfrlem1  6595  tfrlem5  6600  tfrlem9  6605  tfrlem11  6608  oaordex  6760  oaass  6763  odi  6781  omass  6782  oewordri  6794  nnaordex  6840  domtriord  7212  hartogs  7469  card2on  7478  tskwe  7793  infxpenlem  7851  cfub  8085  cfsuc  8093  coflim  8097  hsmexlem2  8263  ondomon  8394  pwcfsdom  8414  inar1  8606  tskord  8611  grudomon  8648  gruina  8649  dfrdg2  25366  poseq  25467  sltres  25532  nobndup  25568  nobnddown  25569  aomclem6  27024
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-uni 3976  df-tr 4263  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545
  Copyright terms: Public domain W3C validator