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Theorem ordsson 6399
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 6392 . 2  |-  Ord  On
2 ordeleqon 6398 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 194 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 465 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 4746 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 232 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 671 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3326   Ord word 4716   Oncon0 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-tr 4384  df-eprel 4630  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721
This theorem is referenced by:  onss  6400  orduni  6403  ordsucuniel  6433  ordsucuni  6438  iordsmo  6816  tfr2b  6853  tz7.44-2  6861  ordiso2  7727  ordtypelem7  7736  ordtypelem8  7737  oiid  7753  r1tr  7981  r1ordg  7983  r1ord3g  7984  r1pwss  7989  r1val1  7991  rankwflemb  7998  r1elwf  8001  rankr1ai  8003  cflim2  8430  cfss  8432  cfslb  8433  cfslbn  8434  cfslb2n  8435  cofsmo  8436  coftr  8440  inaprc  9001  rdgprc  27606  limsucncmpi  28289
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