Theorem domtriord 7660

Description: Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)

Assertion

Ref	Expression
domtriord	|- ( ( A e. On /\ B e. On ) -> ( A ~<_ B <-> -. B ~< A ) )

Proof of Theorem domtriord

Step	Hyp	Ref	Expression
1		sbth 7634	. . . . 5 |- ( ( B ~<_ A /\ A ~<_ B ) -> B ~~ A )
2	1	expcom 435	. . . 4 |- ( A ~<_ B -> ( B ~<_ A -> B ~~ A ) )
3	2	a1i 11	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B -> ( B ~<_ A -> B ~~ A ) ) )
4		iman 424	. . . 4 |- ( ( B ~<_ A -> B ~~ A ) <-> -. ( B ~<_ A /\ -. B ~~ A ) )
5		brsdom 7535	. . . 4 |- ( B ~< A <-> ( B ~<_ A /\ -. B ~~ A ) )
6	4, 5	xchbinxr 311	. . 3 |- ( ( B ~<_ A -> B ~~ A ) <-> -. B ~< A )
7	3, 6	syl6ib 226	. 2 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B -> -. B ~< A ) )
8		onelss 4920	. . . . . . . . . 10 |- ( B e. On -> ( A e. B -> A C_ B ) )
9		ssdomg 7558	. . . . . . . . . 10 |- ( B e. On -> ( A C_ B -> A ~<_ B ) )
10	8, 9	syld 44	. . . . . . . . 9 |- ( B e. On -> ( A e. B -> A ~<_ B ) )
11	10	adantl 466	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( A e. B -> A ~<_ B ) )
12	11	con3d 133	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. A e. B ) )
13		ontri1 4912	. . . . . . . 8 |- ( ( B e. On /\ A e. On ) -> ( B C_ A <-> -. A e. B ) )
14	13	ancoms 453	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B C_ A <-> -. A e. B ) )
15	12, 14	sylibrd 234	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B C_ A ) )
16		ssdomg 7558	. . . . . . 7 |- ( A e. On -> ( B C_ A -> B ~<_ A ) )
17	16	adantr 465	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( B C_ A -> B ~<_ A ) )
18	15, 17	syld 44	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~<_ A ) )
19		ensym 7561	. . . . . . . 8 |- ( B ~~ A -> A ~~ B )
20		endom 7539	. . . . . . . 8 |- ( A ~~ B -> A ~<_ B )
21	19, 20	syl 16	. . . . . . 7 |- ( B ~~ A -> A ~<_ B )
22	21	con3i 135	. . . . . 6 |- ( -. A ~<_ B -> -. B ~~ A )
23	22	a1i 11	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. B ~~ A ) )
24	18, 23	jcad 533	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> ( B ~<_ A /\ -. B ~~ A ) ) )
25	24, 5	syl6ibr 227	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~< A ) )
26	25	con1d 124	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. B ~< A -> A ~<_ B ) )
27	7, 26	impbid 191	1 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B <-> -. B ~< A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   C_ wss 3476   class class class wbr 4447   Oncon0 4878   ~~ cen 7510   ~<_ cdom 7511   ~< csdm 7512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516

This theorem is referenced by:  sdomel  7661  cardsdomel  8351  alephord  8452  alephsucdom  8456  alephdom2  8464