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Theorem onelss 4450
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 4418 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4424 . . 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 1696    C_ wss 3165   Ord word 4407   Oncon0 4408
This theorem is referenced by:  ordunidif  4456  onelssi  4517  ssorduni  4593  suceloni  4620  tfisi  4665  tfrlem1  6407  tfrlem5  6412  tfrlem9  6417  tfrlem11  6420  oaordex  6572  oaass  6575  odi  6593  omass  6594  oewordri  6606  nnaordex  6652  domtriord  7023  hartogs  7275  card2on  7284  tskwe  7599  infxpenlem  7657  cfub  7891  cfsuc  7899  coflim  7903  hsmexlem2  8069  ondomon  8201  pwcfsdom  8221  inar1  8413  tskord  8418  grudomon  8455  gruina  8456  dfrdg2  24223  poseq  24324  sltres  24389  nobndup  24425  nobnddown  24426  aomclem6  27259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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