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Theorem ordsson 4712
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 4705 . 2  |-  Ord  On
2 ordeleqon 4711 . . . . 5  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
32biimpi 187 . . . 4  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
43adantr 452 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On ) )
5 ordsseleq 4553 . . 3  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  C_  On  <->  ( A  e.  On  \/  A  =  On ) ) )
64, 5mpbird 224 . 2  |-  ( ( Ord  A  /\  Ord  On )  ->  A  C_  On )
71, 6mpan2 653 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   Ord word 4523   Oncon0 4524
This theorem is referenced by:  onss  4713  orduni  4716  ordsucuniel  4746  ordsucuni  4751  iordsmo  6557  tfr2b  6595  tz7.44-2  6603  ordiso2  7419  ordtypelem7  7428  ordtypelem8  7429  oiid  7445  r1tr  7637  r1ordg  7639  r1ord3g  7640  r1pwss  7645  r1val1  7647  rankwflemb  7654  r1elwf  7657  rankr1ai  7659  cflim2  8078  cfss  8080  cfslb  8081  cfslbn  8082  cfslb2n  8083  cofsmo  8084  coftr  8088  inaprc  8646  rdgprc  25177  tfrALTlem  25302  limsucncmpi  25911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-tr 4246  df-eprel 4437  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528
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