Theorem ordsson 4236

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 4138	. . 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 3005	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 1433   C_ wss 2973   Ord word 4117   Oncon0 4118

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  onss  4237  orduni  4239  iordsmo  5935  tfrlemi14d  5970  ordiso2  6446