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Theorem onelss 4910
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 4878 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4884 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
32ex 434 . 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 setvar class
Syntax hints:    -> wi 4    e. wcel 1804    C_ wss 3461   Ord word 4867   Oncon0 4868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-v 3097  df-in 3468  df-ss 3475  df-uni 4235  df-tr 4531  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872
This theorem is referenced by:  ordunidif  4916  onelssi  4976  ssorduni  6606  suceloni  6633  tfisi  6678  tfrlem9  7056  tfrlem11  7059  oaordex  7209  oaass  7212  odi  7230  omass  7231  oewordri  7243  nnaordex  7289  domtriord  7665  hartogs  7972  card2on  7983  tskwe  8334  infxpenlem  8394  cfub  8632  cfsuc  8640  coflim  8644  hsmexlem2  8810  ondomon  8941  pwcfsdom  8961  inar1  9156  tskord  9161  grudomon  9198  gruina  9199  dfrdg2  29203  poseq  29308  sltres  29399  nobndup  29435  nobnddown  29436  aomclem6  30980
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