Theorem ordelon 4343

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 4341	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4333	. . 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 2128   Ord word 4322   Oncon0 4323

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328

This theorem is referenced by:  onelon  4344  ordsson  4451  ordpwsucss  4526  tfr1onlemsucfn  6287  tfr1onlemsucaccv  6288  tfr1onlembfn  6291  tfr1onlemubacc  6293  tfr1onlemaccex  6295  tfrcllemsucfn  6300  tfrcllemsucaccv  6301  tfrcllembfn  6304  tfrcllemubacc  6306  tfrcllemaccex  6308  tfrcl  6311