Theorem ordelord 4298

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)

Assertion

Ref	Expression
ordelord	|- ( ( Ord A /\ B e. A ) -> Ord B )

Proof of Theorem ordelord

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2200	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	anbi2d 459	. . . 4 |- ( x = B -> ( ( Ord A /\ x e. A ) <-> ( Ord A /\ B e. A ) ) )
3		ordeq 4289	. . . 4 |- ( x = B -> ( Ord x <-> Ord B ) )
4	2, 3	imbi12d 233	. . 3 |- ( x = B -> ( ( ( Ord A /\ x e. A ) -> Ord x ) <-> ( ( Ord A /\ B e. A ) -> Ord B ) ) )
5		dford3 4284	. . . . . 6 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
6	5	simprbi 273	. . . . 5 |- ( Ord A -> A. x e. A Tr x )
7	6	r19.21bi 2518	. . . 4 |- ( ( Ord A /\ x e. A ) -> Tr x )
8		ordelss 4296	. . . 4 |- ( ( Ord A /\ x e. A ) -> x C_ A )
9		simpl 108	. . . 4 |- ( ( Ord A /\ x e. A ) -> Ord A )
10		trssord 4297	. . . 4 |- ( ( Tr x /\ x C_ A /\ Ord A ) -> Ord x )
11	7, 8, 9, 10	syl3anc 1216	. . 3 |- ( ( Ord A /\ x e. A ) -> Ord x )
12	4, 11	vtoclg 2741	. 2 |- ( B e. A -> ( ( Ord A /\ B e. A ) -> Ord B ) )
13	12	anabsi7 570	1 |- ( ( Ord A /\ B e. A ) -> Ord B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2414   C_ wss 3066   Tr wtr 4021   Ord word 4279

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283

This theorem is referenced by:  tron  4299  ordelon  4300  ordsucg  4413  ordwe  4485  smores  6182