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Theorem ordsson 4518
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 4511 . 2  |-  Ord  On
2 ordeleqon 4517 . . . . 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 4358 . . 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 3094   Ord word 4328   Oncon0 4329
This theorem is referenced by:  onss  4519  orduni  4522  ordsucuniel  4552  ordsucuni  4557  iordsmo  6307  tfr2b  6345  tz7.44-2  6353  ordiso2  7163  ordtypelem7  7172  ordtypelem8  7173  oiid  7189  r1tr  7381  r1ordg  7383  r1ord3g  7384  r1pwss  7389  r1val1  7391  rankwflemb  7398  r1elwf  7401  rankr1ai  7403  cflim2  7822  cfss  7824  cfslb  7825  cfslbn  7826  cfslb2n  7827  cofsmo  7828  coftr  7832  inaprc  8391  rdgprc  23485  tfrALTlem  23610  limsucncmpi  24224
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333
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