Theorem ordelon 4305

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 4303	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4295	. . 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 4284   Oncon0 4285

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 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  onelon  4306  ordsson  4408  ordpwsucss  4482  tfr1onlemsucfn  6237  tfr1onlemsucaccv  6238  tfr1onlembfn  6241  tfr1onlemubacc  6243  tfr1onlemaccex  6245  tfrcllemsucfn  6250  tfrcllemsucaccv  6251  tfrcllembfn  6254  tfrcllemubacc  6256  tfrcllemaccex  6258  tfrcl  6261