Theorem domtriord 7682

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 7656	. . . . 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 7557	. . . 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 4929	. . . . . . . . . 10 |- ( B e. On -> ( A e. B -> A C_ B ) )
9		ssdomg 7580	. . . . . . . . . 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 4921	. . . . . . . 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 7580	. . . . . . 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 7583	. . . . . . . 8 |- ( B ~~ A -> A ~~ B )
20		endom 7561	. . . . . . . 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 1819   C_ wss 3471   class class class wbr 4456   Oncon0 4887   ~~ cen 7532   ~<_ cdom 7533   ~< csdm 7534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538

This theorem is referenced by:  sdomel  7683  cardsdomel  8372  alephord  8473  alephsucdom  8477  alephdom2  8485