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Theorem onelss 4434
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 4402 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4408 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
32ex 423 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  B  C_  A ) )
41, 3syl 15 1  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ordunidif  4440  onelssi  4501  ssorduni  4577  suceloni  4604  tfisi  4649  tfrlem1  6391  tfrlem5  6396  tfrlem9  6401  tfrlem11  6404  oaordex  6556  oaass  6559  odi  6577  omass  6578  oewordri  6590  nnaordex  6636  domtriord  7007  hartogs  7259  card2on  7268  tskwe  7583  infxpenlem  7641  cfub  7875  cfsuc  7883  coflim  7887  hsmexlem2  8053  ondomon  8185  pwcfsdom  8205  inar1  8397  tskord  8402  grudomon  8439  gruina  8440  dfrdg2  24152  poseq  24253  sltres  24318  nobndup  24354  nobnddown  24355  aomclem6  27156
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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