Theorem ordelon 4419

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 4417	. 2 |- ( ( Ord A /\ B e. A ) -> Ord B )
2		elong 4409	. . 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 2167   Ord word 4398   Oncon0 4399

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404

This theorem is referenced by:  onelon  4420  ordsson  4529  ordpwsucss  4604  tfr1onlemsucfn  6407  tfr1onlemsucaccv  6408  tfr1onlembfn  6411  tfr1onlemubacc  6413  tfr1onlemaccex  6415  tfrcllemsucfn  6420  tfrcllemsucaccv  6421  tfrcllembfn  6424  tfrcllemubacc  6426  tfrcllemaccex  6428  tfrcl  6431