Theorem ordelord 4429

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 2268	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	anbi2d 464	. . . 4 |- ( x = B -> ( ( Ord A /\ x e. A ) <-> ( Ord A /\ B e. A ) ) )
3		ordeq 4420	. . . 4 |- ( x = B -> ( Ord x <-> Ord B ) )
4	2, 3	imbi12d 234	. . 3 |- ( x = B -> ( ( ( Ord A /\ x e. A ) -> Ord x ) <-> ( ( Ord A /\ B e. A ) -> Ord B ) ) )
5		dford3 4415	. . . . . 6 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
6	5	simprbi 275	. . . . 5 |- ( Ord A -> A. x e. A Tr x )
7	6	r19.21bi 2594	. . . 4 |- ( ( Ord A /\ x e. A ) -> Tr x )
8		ordelss 4427	. . . 4 |- ( ( Ord A /\ x e. A ) -> x C_ A )
9		simpl 109	. . . 4 |- ( ( Ord A /\ x e. A ) -> Ord A )
10		trssord 4428	. . . 4 |- ( ( Tr x /\ x C_ A /\ Ord A ) -> Ord x )
11	7, 8, 9, 10	syl3anc 1250	. . 3 |- ( ( Ord A /\ x e. A ) -> Ord x )
12	4, 11	vtoclg 2833	. 2 |- ( B e. A -> ( ( Ord A /\ B e. A ) -> Ord B ) )
13	12	anabsi7 581	1 |- ( ( Ord A /\ B e. A ) -> Ord B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   Tr wtr 4143   Ord word 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4144  df-iord 4414

This theorem is referenced by:  tron  4430  ordelon  4431  ordsucg  4551  ordwe  4625  smores  6380