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Theorem ordsson 6574
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 6567 . 2  |-  Ord  On
2 ordeleqon 6573 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 197 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 466 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 5414 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 235 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 675 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3379   Ord word 5384   Oncon0 5385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389
This theorem is referenced by:  onss  6575  orduni  6579  ordsucuniel  6609  ordsucuni  6614  iordsmo  7031  dfrecs3  7046  tfr2b  7069  tz7.44-2  7080  ordiso2  7983  ordtypelem7  7992  ordtypelem8  7993  oiid  8009  r1tr  8199  r1ordg  8201  r1ord3g  8202  r1pwss  8207  r1val1  8209  rankwflemb  8216  r1elwf  8219  rankr1ai  8221  cflim2  8644  cfss  8646  cfslb  8647  cfslbn  8648  cfslb2n  8649  cofsmo  8650  coftr  8654  inaprc  9212  rdgprc  30392  limsucncmpi  31054
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