Theorem domtriord 8106

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 8080	. . . . 5 |- ( ( B ~<_ A /\ A ~<_ B ) -> B ~~ A )
2	1	expcom 451	. . . 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 440	. . . 4 |- ( ( B ~<_ A -> B ~~ A ) <-> -. ( B ~<_ A /\ -. B ~~ A ) )
5		brsdom 7978	. . . 4 |- ( B ~< A <-> ( B ~<_ A /\ -. B ~~ A ) )
6	4, 5	xchbinxr 325	. . 3 |- ( ( B ~<_ A -> B ~~ A ) <-> -. B ~< A )
7	3, 6	syl6ib 241	. 2 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B -> -. B ~< A ) )
8		onelss 5766	. . . . . . . . . 10 |- ( B e. On -> ( A e. B -> A C_ B ) )
9		ssdomg 8001	. . . . . . . . . 10 |- ( B e. On -> ( A C_ B -> A ~<_ B ) )
10	8, 9	syld 47	. . . . . . . . 9 |- ( B e. On -> ( A e. B -> A ~<_ B ) )
11	10	adantl 482	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( A e. B -> A ~<_ B ) )
12	11	con3d 148	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. A e. B ) )
13		ontri1 5757	. . . . . . . 8 |- ( ( B e. On /\ A e. On ) -> ( B C_ A <-> -. A e. B ) )
14	13	ancoms 469	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B C_ A <-> -. A e. B ) )
15	12, 14	sylibrd 249	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B C_ A ) )
16		ssdomg 8001	. . . . . . 7 |- ( A e. On -> ( B C_ A -> B ~<_ A ) )
17	16	adantr 481	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( B C_ A -> B ~<_ A ) )
18	15, 17	syld 47	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~<_ A ) )
19		ensym 8005	. . . . . . . 8 |- ( B ~~ A -> A ~~ B )
20		endom 7982	. . . . . . . 8 |- ( A ~~ B -> A ~<_ B )
21	19, 20	syl 17	. . . . . . 7 |- ( B ~~ A -> A ~<_ B )
22	21	con3i 150	. . . . . 6 |- ( -. A ~<_ B -> -. B ~~ A )
23	22	a1i 11	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. B ~~ A ) )
24	18, 23	jcad 555	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> ( B ~<_ A /\ -. B ~~ A ) ) )
25	24, 5	syl6ibr 242	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~< A ) )
26	25	con1d 139	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. B ~< A -> A ~<_ B ) )
27	7, 26	impbid 202	1 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B <-> -. B ~< A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   class class class wbr 4653   Oncon0 5723   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  sdomel  8107  cardsdomel  8800  alephord  8898  alephsucdom  8902  alephdom2  8910