Theorem ordelon 4361

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 4359	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4351	. . 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 2136   Ord word 4340   Oncon0 4341

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  onelon  4362  ordsson  4469  ordpwsucss  4544  tfr1onlemsucfn  6308  tfr1onlemsucaccv  6309  tfr1onlembfn  6312  tfr1onlemubacc  6314  tfr1onlemaccex  6316  tfrcllemsucfn  6321  tfrcllemsucaccv  6322  tfrcllembfn  6325  tfrcllemubacc  6327  tfrcllemaccex  6329  tfrcl  6332