Theorem ordelord 4311

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 2203	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	anbi2d 460	. . . 4 |- ( x = B -> ( ( Ord A /\ x e. A ) <-> ( Ord A /\ B e. A ) ) )
3		ordeq 4302	. . . 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 4297	. . . . . 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 2523	. . . 4 |- ( ( Ord A /\ x e. A ) -> Tr x )
8		ordelss 4309	. . . 4 |- ( ( Ord A /\ x e. A ) -> x C_ A )
9		simpl 108	. . . 4 |- ( ( Ord A /\ x e. A ) -> Ord A )
10		trssord 4310	. . . 4 |- ( ( Tr x /\ x C_ A /\ Ord A ) -> Ord x )
11	7, 8, 9, 10	syl3anc 1217	. . 3 |- ( ( Ord A /\ x e. A ) -> Ord x )
12	4, 11	vtoclg 2749	. 2 |- ( B e. A -> ( ( Ord A /\ B e. A ) -> Ord B ) )
13	12	anabsi7 571	1 |- ( ( Ord A /\ B e. A ) -> Ord B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417   C_ wss 3076   Tr wtr 4034   Ord word 4292

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296

This theorem is referenced by:  tron  4312  ordelon  4313  ordsucg  4426  ordwe  4498  smores  6197