Theorem ordelon 4167

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 4165	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4157	. . 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 1434   Ord word 4146   Oncon0 4147

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 2613  df-in 2989  df-ss 2996  df-uni 3623  df-tr 3897  df-iord 4150  df-on 4152

This theorem is referenced by:  onelon  4168  ordsson  4265  ordpwsucss  4339  tfr1onlemsucfn  6011  tfr1onlemsucaccv  6012  tfr1onlembfn  6015  tfr1onlemubacc  6017  tfr1onlemaccex  6019  tfrcllemsucfn  6024  tfrcllemsucaccv  6025  tfrcllembfn  6028  tfrcllemubacc  6030  tfrcllemaccex  6032  tfrcl  6035