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Theorem ordsson 4737
Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordsson  |-  ( Ord 
A  ->  A  C_  On )

Proof of Theorem ordsson
StepHypRef Expression
1 ordon 4730 . 2  |-  Ord  On
2 ordeleqon 4736 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 187 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 452 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 4578 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 224 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 653 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3288   Ord word 4548   Oncon0 4549
This theorem is referenced by:  onss  4738  orduni  4741  ordsucuniel  4771  ordsucuni  4776  iordsmo  6586  tfr2b  6624  tz7.44-2  6632  ordiso2  7448  ordtypelem7  7457  ordtypelem8  7458  oiid  7474  r1tr  7666  r1ordg  7668  r1ord3g  7669  r1pwss  7674  r1val1  7676  rankwflemb  7683  r1elwf  7686  rankr1ai  7688  cflim2  8107  cfss  8109  cfslb  8110  cfslbn  8111  cfslb2n  8112  cofsmo  8113  coftr  8117  inaprc  8675  rdgprc  25373  tfrALTlem  25498  limsucncmpi  26107
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553
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