Theorem ordsson 4271

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.

Step	Hyp	Ref	Expression
1		ordelon 4173	. . 3 |- ( ( Ord A /\ x e. A ) -> x e. On )
2	1	ex 113	. 2 |- ( Ord A -> ( x e. A -> x e. On ) )
3	2	ssrdv 3016	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 1434   C_ wss 2984   Ord word 4152   Oncon0 4153

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158

This theorem is referenced by:  onss  4272  orduni  4274  iordsmo  5992  tfrlemi14d  6028  tfr1onlemssrecs  6034  tfri1dALT  6046  tfrcllemssrecs  6047  ordiso2  6633