Theorem ordelon 4380

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 4378	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4370	. . 3 |- ( B e. A -> ( B e. On <-> Ord B ) )
3	2	adantl 277	. 2 |- ( ( Ord A /\ B e. A ) -> ( B e. On <-> Ord B ) )
4	1, 3	mpbird 167	1 |- ( ( Ord A /\ B e. A ) -> B e. On )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   Ord word 4359   Oncon0 4360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365

This theorem is referenced by:  onelon  4381  ordsson  4488  ordpwsucss  4563  tfr1onlemsucfn  6335  tfr1onlemsucaccv  6336  tfr1onlembfn  6339  tfr1onlemubacc  6341  tfr1onlemaccex  6343  tfrcllemsucfn  6348  tfrcllemsucaccv  6349  tfrcllembfn  6352  tfrcllemubacc  6354  tfrcllemaccex  6356  tfrcl  6359