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Theorem ordsson 4472
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 4465 . 2  |-  Ord  On
2 ordeleqon 4471 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 188 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 453 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 4314 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 225 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 655 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3078   Ord word 4284   Oncon0 4285
This theorem is referenced by:  onss  4473  orduni  4476  ordsucuniel  4506  ordsucuni  4511  iordsmo  6260  tfr2b  6298  tz7.44-2  6306  ordiso2  7114  ordtypelem7  7123  ordtypelem8  7124  oiid  7140  r1tr  7332  r1ordg  7334  r1ord3g  7335  r1pwss  7340  r1val1  7342  rankwflemb  7349  r1elwf  7352  rankr1ai  7354  cflim2  7773  cfss  7775  cfslb  7776  cfslbn  7777  cfslb2n  7778  cofsmo  7779  coftr  7783  inaprc  8338  rdgprc  23319  tfrALTlem  23444  limsucncmpi  24058
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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