Theorem ordelord 4118

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 2100	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	anbi2d 437	. . . 4 |- ( x = B -> ( ( Ord A /\ x e. A ) <-> ( Ord A /\ B e. A ) ) )
3		ordeq 4109	. . . 4 |- ( x = B -> ( Ord x <-> Ord B ) )
4	2, 3	imbi12d 223	. . 3 |- ( x = B -> ( ( ( Ord A /\ x e. A ) -> Ord x ) <-> ( ( Ord A /\ B e. A ) -> Ord B ) ) )
5		dford3 4104	. . . . . 6 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
6	5	simprbi 260	. . . . 5 |- ( Ord A -> A. x e. A Tr x )
7	6	r19.21bi 2407	. . . 4 |- ( ( Ord A /\ x e. A ) -> Tr x )
8		ordelss 4116	. . . 4 |- ( ( Ord A /\ x e. A ) -> x C_ A )
9		simpl 102	. . . 4 |- ( ( Ord A /\ x e. A ) -> Ord A )
10		trssord 4117	. . . 4 |- ( ( Tr x /\ x C_ A /\ Ord A ) -> Ord x )
11	7, 8, 9, 10	syl3anc 1135	. . 3 |- ( ( Ord A /\ x e. A ) -> Ord x )
12	4, 11	vtoclg 2613	. 2 |- ( B e. A -> ( ( Ord A /\ B e. A ) -> Ord B ) )
13	12	anabsi7 515	1 |- ( ( Ord A /\ B e. A ) -> Ord B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   C_ wss 2917   Tr wtr 3854   Ord word 4099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103

This theorem is referenced by:  tron  4119  ordelon  4120  ordsucg  4228  ordwe  4300  smores  5907