Theorem ordelon 4430

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 4428	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4420	. . 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 2176   Ord word 4409   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  onelon  4431  ordsson  4540  ordpwsucss  4615  tfr1onlemsucfn  6426  tfr1onlemsucaccv  6427  tfr1onlembfn  6430  tfr1onlemubacc  6432  tfr1onlemaccex  6434  tfrcllemsucfn  6439  tfrcllemsucaccv  6440  tfrcllembfn  6443  tfrcllemubacc  6445  tfrcllemaccex  6447  tfrcl  6450