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Theorem ordsson 4309
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.)
Assertion
Ref Expression
ordsson  |-  ( Ord 
A  ->  A  C_  On )

Proof of Theorem ordsson
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordelon 4210 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  e.  On )
21ex 113 . 2  |-  ( Ord 
A  ->  ( x  e.  A  ->  x  e.  On ) )
32ssrdv 3031 1  |-  ( Ord 
A  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438    C_ wss 2999   Ord word 4189   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  onss  4310  orduni  4312  iordsmo  6062  tfrlemi14d  6098  tfr1onlemssrecs  6104  tfri1dALT  6116  tfrcllemssrecs  6117  ordiso2  6728
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