Theorem ordelord 4208

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 2150	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	anbi2d 452	. . . 4 |- ( x = B -> ( ( Ord A /\ x e. A ) <-> ( Ord A /\ B e. A ) ) )
3		ordeq 4199	. . . 4 |- ( x = B -> ( Ord x <-> Ord B ) )
4	2, 3	imbi12d 232	. . 3 |- ( x = B -> ( ( ( Ord A /\ x e. A ) -> Ord x ) <-> ( ( Ord A /\ B e. A ) -> Ord B ) ) )
5		dford3 4194	. . . . . 6 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
6	5	simprbi 269	. . . . 5 |- ( Ord A -> A. x e. A Tr x )
7	6	r19.21bi 2461	. . . 4 |- ( ( Ord A /\ x e. A ) -> Tr x )
8		ordelss 4206	. . . 4 |- ( ( Ord A /\ x e. A ) -> x C_ A )
9		simpl 107	. . . 4 |- ( ( Ord A /\ x e. A ) -> Ord A )
10		trssord 4207	. . . 4 |- ( ( Tr x /\ x C_ A /\ Ord A ) -> Ord x )
11	7, 8, 9, 10	syl3anc 1174	. . 3 |- ( ( Ord A /\ x e. A ) -> Ord x )
12	4, 11	vtoclg 2679	. 2 |- ( B e. A -> ( ( Ord A /\ B e. A ) -> Ord B ) )
13	12	anabsi7 548	1 |- ( ( Ord A /\ B e. A ) -> Ord B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   A.wral 2359   C_ wss 2999   Tr wtr 3936   Ord word 4189

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193

This theorem is referenced by:  tron  4209  ordelon  4210  ordsucg  4319  ordwe  4391  smores  6057