Theorem domtriord 7557

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 7531	. . . . 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 7432	. . . 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 4859	. . . . . . . . . 10 |- ( B e. On -> ( A e. B -> A C_ B ) )
9		ssdomg 7455	. . . . . . . . . 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 4851	. . . . . . . 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 7455	. . . . . . 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 7458	. . . . . . . 8 |- ( B ~~ A -> A ~~ B )
20		endom 7436	. . . . . . . 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 1758   C_ wss 3426   class class class wbr 4390   Oncon0 4817   ~~ cen 7407   ~<_ cdom 7408   ~< csdm 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413

This theorem is referenced by:  sdomel  7558  cardsdomel  8245  alephord  8346  alephsucdom  8350  alephdom2  8358