Theorem domtriord 7721

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 7695	. . . . 5 |- ( ( B ~<_ A /\ A ~<_ B ) -> B ~~ A )
2	1	expcom 436	. . . 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 425	. . . 4 |- ( ( B ~<_ A -> B ~~ A ) <-> -. ( B ~<_ A /\ -. B ~~ A ) )
5		brsdom 7596	. . . 4 |- ( B ~< A <-> ( B ~<_ A /\ -. B ~~ A ) )
6	4, 5	xchbinxr 312	. . 3 |- ( ( B ~<_ A -> B ~~ A ) <-> -. B ~< A )
7	3, 6	syl6ib 229	. 2 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B -> -. B ~< A ) )
8		onelss 5481	. . . . . . . . . 10 |- ( B e. On -> ( A e. B -> A C_ B ) )
9		ssdomg 7619	. . . . . . . . . 10 |- ( B e. On -> ( A C_ B -> A ~<_ B ) )
10	8, 9	syld 45	. . . . . . . . 9 |- ( B e. On -> ( A e. B -> A ~<_ B ) )
11	10	adantl 467	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( A e. B -> A ~<_ B ) )
12	11	con3d 138	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. A e. B ) )
13		ontri1 5473	. . . . . . . 8 |- ( ( B e. On /\ A e. On ) -> ( B C_ A <-> -. A e. B ) )
14	13	ancoms 454	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( B C_ A <-> -. A e. B ) )
15	12, 14	sylibrd 237	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B C_ A ) )
16		ssdomg 7619	. . . . . . 7 |- ( A e. On -> ( B C_ A -> B ~<_ A ) )
17	16	adantr 466	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( B C_ A -> B ~<_ A ) )
18	15, 17	syld 45	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~<_ A ) )
19		ensym 7622	. . . . . . . 8 |- ( B ~~ A -> A ~~ B )
20		endom 7600	. . . . . . . 8 |- ( A ~~ B -> A ~<_ B )
21	19, 20	syl 17	. . . . . . 7 |- ( B ~~ A -> A ~<_ B )
22	21	con3i 140	. . . . . 6 |- ( -. A ~<_ B -> -. B ~~ A )
23	22	a1i 11	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> -. B ~~ A ) )
24	18, 23	jcad 535	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> ( B ~<_ A /\ -. B ~~ A ) ) )
25	24, 5	syl6ibr 230	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A ~<_ B -> B ~< A ) )
26	25	con1d 127	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. B ~< A -> A ~<_ B ) )
27	7, 26	impbid 193	1 |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B <-> -. B ~< A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   e. wcel 1868   C_ wss 3436   class class class wbr 4420   Oncon0 5439   ~~ cen 7571   ~<_ cdom 7572   ~< csdm 7573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-ord 5442  df-on 5443  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577

This theorem is referenced by:  sdomel  7722  cardsdomel  8410  alephord  8507  alephsucdom  8511  alephdom2  8519