Theorem ordelon 4300

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 4298	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4290	. . 3 |- ( B e. A -> ( B e. On <-> Ord B ) )
3	2	adantl 275	. 2 |- ( ( Ord A /\ B e. A ) -> ( B e. On <-> Ord B ) )
4	1, 3	mpbird 166	1 |- ( ( Ord A /\ B e. A ) -> B e. On )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   Ord word 4279   Oncon0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285

This theorem is referenced by:  onelon  4301  ordsson  4403  ordpwsucss  4477  tfr1onlemsucfn  6230  tfr1onlemsucaccv  6231  tfr1onlembfn  6234  tfr1onlemubacc  6236  tfr1onlemaccex  6238  tfrcllemsucfn  6243  tfrcllemsucaccv  6244  tfrcllembfn  6247  tfrcllemubacc  6249  tfrcllemaccex  6251  tfrcl  6254