Theorem domtriord 5546

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 5520	. . . . 5 |- ((B ~<_ A /\ A ~<_ B) -> B ~~ A)
2	1	expcom 403	. . . 4 |- (A ~<_ B -> (B ~<_ A -> B ~~ A))
3	2	a1i 8	. . 3 |- ((A e. On /\ B e. On) -> (A ~<_ B -> (B ~<_ A -> B ~~ A)))
4		iman 256	. . . 4 |- ((B ~<_ A -> B ~~ A) <-> -. (B ~<_ A /\ -. B ~~ A))
5		brsdom 5440	. . . . . 6 |- (B ~< A <-> (B ~<_ A /\ -. B ~~ A))
6	5	notbii 204	. . . . 5 |- (-. B ~< A <-> -. (B ~<_ A /\ -. B ~~ A))
7	6	bicomi 189	. . . 4 |- (-. (B ~<_ A /\ -. B ~~ A) <-> -. B ~< A)
8	4, 7	bitri 190	. . 3 |- ((B ~<_ A -> B ~~ A) <-> -. B ~< A)
9	3, 8	syl6ib 229	. 2 |- ((A e. On /\ B e. On) -> (A ~<_ B -> -. B ~< A))
10		onelss 3705	. . . . . . . . 9 |- (B e. On -> (A e. B -> A C_ B))
11		ssdomg 5467	. . . . . . . . 9 |- (A e. On -> (A C_ B -> A ~<_ B))
12	10, 11	sylan9r 519	. . . . . . . 8 |- ((A e. On /\ B e. On) -> (A e. B -> A ~<_ B))
13	12	con3d 111	. . . . . . 7 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> -. A e. B))
14		ordtri1 3693	. . . . . . . . 9 |- ((Ord B /\ Ord A) -> (B C_ A <-> -. A e. B))
15		eloni 3667	. . . . . . . . 9 |- (B e. On -> Ord B)
16		eloni 3667	. . . . . . . . 9 |- (A e. On -> Ord A)
17	14, 15, 16	syl2an 503	. . . . . . . 8 |- ((B e. On /\ A e. On) -> (B C_ A <-> -. A e. B))
18	17	ancoms 484	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B C_ A <-> -. A e. B))
19	13, 18	sylibrd 221	. . . . . 6 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> B C_ A))
20		ssdomg 5467	. . . . . . 7 |- (B e. On -> (B C_ A -> B ~<_ A))
21	20	adantl 424	. . . . . 6 |- ((A e. On /\ B e. On) -> (B C_ A -> B ~<_ A))
22	19, 21	syld 30	. . . . 5 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> B ~<_ A))
23		ensymg 5470	. . . . . . . 8 |- (A e. On -> (B ~~ A -> A ~~ B))
24	23	adantr 425	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B ~~ A -> A ~~ B))
25		endom 5444	. . . . . . 7 |- (A ~~ B -> A ~<_ B)
26	24, 25	syl6 25	. . . . . 6 |- ((A e. On /\ B e. On) -> (B ~~ A -> A ~<_ B))
27	26	con3d 111	. . . . 5 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> -. B ~~ A))
28	22, 27	jcad 661	. . . 4 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> (B ~<_ A /\ -. B ~~ A)))
29	28, 5	syl6ibr 230	. . 3 |- ((A e. On /\ B e. On) -> (-. A ~<_ B -> B ~< A))
30	29	con1d 109	. 2 |- ((A e. On /\ B e. On) -> (-. B ~< A -> A ~<_ B))
31	9, 30	impbid 574	1 |- ((A e. On /\ B e. On) -> (A ~<_ B <-> -. B ~< A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300   C_ wss 2593   class class class wbr 3338  Ord word 3656  Oncon0 3657   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  omsubsuc2 5878  elomsubsd 5885  omsubdmss 5886  omsubsuc2OLD 15387  elomsubsdOLD 15394  omsubdmssOLD 15395

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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