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Theorem onelss 4623
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 4591 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4597 . . 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 1725    C_ wss 3320   Ord word 4580   Oncon0 4581
This theorem is referenced by:  ordunidif  4629  onelssi  4690  ssorduni  4766  suceloni  4793  tfisi  4838  tfrlem1  6636  tfrlem5  6641  tfrlem9  6646  tfrlem11  6649  oaordex  6801  oaass  6804  odi  6822  omass  6823  oewordri  6835  nnaordex  6881  domtriord  7253  hartogs  7513  card2on  7522  tskwe  7837  infxpenlem  7895  cfub  8129  cfsuc  8137  coflim  8141  hsmexlem2  8307  ondomon  8438  pwcfsdom  8458  inar1  8650  tskord  8655  grudomon  8692  gruina  8693  dfrdg2  25423  poseq  25528  sltres  25619  nobndup  25655  nobnddown  25656  aomclem6  27134
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-uni 4016  df-tr 4303  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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