Theorem ordelon 4138

Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
ordelon	|- ( ( Ord A /\ B e. A ) -> B e. On )

Proof of Theorem ordelon

Step	Hyp	Ref	Expression
1		ordelord 4136	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4128	. . 3 |- ( B e. A -> ( B e. On <-> Ord B ) )
3	2	adantl 271	. 2 |- ( ( Ord A /\ B e. A ) -> ( B e. On <-> Ord B ) )
4	1, 3	mpbird 165	1 |- ( ( Ord A /\ B e. A ) -> B e. On )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   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:  onelon  4139  ordsson  4236  ordpwsucss  4310