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Theorem ordsson 4597
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 4590 . 2  |-  Ord  On
2 ordeleqon 4596 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 186 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 451 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 4437 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 223 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 652 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   Ord word 4407   Oncon0 4408
This theorem is referenced by:  onss  4598  orduni  4601  ordsucuniel  4631  ordsucuni  4636  iordsmo  6390  tfr2b  6428  tz7.44-2  6436  ordiso2  7246  ordtypelem7  7255  ordtypelem8  7256  oiid  7272  r1tr  7464  r1ordg  7466  r1ord3g  7467  r1pwss  7472  r1val1  7474  rankwflemb  7481  r1elwf  7484  rankr1ai  7486  cflim2  7905  cfss  7907  cfslb  7908  cfslbn  7909  cfslb2n  7910  cofsmo  7911  coftr  7915  inaprc  8474  rdgprc  24222  tfrALTlem  24347  limsucncmpi  24956
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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