Theorem ordelon 4368

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 4366	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4358	. . 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 2141   Ord word 4347   Oncon0 4348

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  onelon  4369  ordsson  4476  ordpwsucss  4551  tfr1onlemsucfn  6319  tfr1onlemsucaccv  6320  tfr1onlembfn  6323  tfr1onlemubacc  6325  tfr1onlemaccex  6327  tfrcllemsucfn  6332  tfrcllemsucaccv  6333  tfrcllembfn  6336  tfrcllemubacc  6338  tfrcllemaccex  6340  tfrcl  6343