Theorem swoord1 6451

Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypotheses

Ref	Expression
swoer.1	|- R = ( ( X X. X ) \ ( .< u. `' .< ) )
swoer.2	|- ( ( ph /\ ( y e. X /\ z e. X ) ) -> ( y .< z -> -. z .< y ) )
swoer.3	|- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )
swoord.4	|- ( ph -> B e. X )
swoord.5	|- ( ph -> C e. X )
swoord.6	|- ( ph -> A R B )

Assertion

Ref	Expression
swoord1	|- ( ph -> ( A .< C <-> B .< C ) )

Distinct variable groups:   x, y, z, .<   x, A, y, z   x, B, y, z   x, C, y, z   ph, x, y, z   x, X, y, z

Allowed substitution hints:   R( x, y, z)

Proof of Theorem swoord1

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( ph -> ph )
2		swoord.6	. . . . 5 |- ( ph -> A R B )
3		swoer.1	. . . . . . 7 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
4		difss 3197	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
5	3, 4	eqsstri 3124	. . . . . 6 |- R C_ ( X X. X )
6	5	ssbri 3967	. . . . 5 |- ( A R B -> A ( X X. X ) B )
7		df-br 3925	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
8		opelxp1 4568	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
9	7, 8	sylbi 120	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
10	2, 6, 9	3syl 17	. . . 4 |- ( ph -> A e. X )
11		swoord.5	. . . 4 |- ( ph -> C e. X )
12		swoord.4	. . . 4 |- ( ph -> B e. X )
13		swoer.3	. . . . 5 |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )
14	13	swopolem 4222	. . . 4 |- ( ( ph /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( A .< C -> ( A .< B \/ B .< C ) ) )
15	1, 10, 11, 12, 14	syl13anc 1218	. . 3 |- ( ph -> ( A .< C -> ( A .< B \/ B .< C ) ) )
16	3	brdifun 6449	. . . . . . 7 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
17	10, 12, 16	syl2anc 408	. . . . . 6 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	2, 17	mpbid 146	. . . . 5 |- ( ph -> -. ( A .< B \/ B .< A ) )
19		orc 701	. . . . 5 |- ( A .< B -> ( A .< B \/ B .< A ) )
20	18, 19	nsyl 617	. . . 4 |- ( ph -> -. A .< B )
21		biorf 733	. . . 4 |- ( -. A .< B -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
22	20, 21	syl 14	. . 3 |- ( ph -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
23	15, 22	sylibrd 168	. 2 |- ( ph -> ( A .< C -> B .< C ) )
24	13	swopolem 4222	. . . 4 |- ( ( ph /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( B .< C -> ( B .< A \/ A .< C ) ) )
25	1, 12, 11, 10, 24	syl13anc 1218	. . 3 |- ( ph -> ( B .< C -> ( B .< A \/ A .< C ) ) )
26		olc 700	. . . . 5 |- ( B .< A -> ( A .< B \/ B .< A ) )
27	18, 26	nsyl 617	. . . 4 |- ( ph -> -. B .< A )
28		biorf 733	. . . 4 |- ( -. B .< A -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
29	27, 28	syl 14	. . 3 |- ( ph -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
30	25, 29	sylibrd 168	. 2 |- ( ph -> ( B .< C -> A .< C ) )
31	23, 30	impbid 128	1 |- ( ph -> ( A .< C <-> B .< C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   \ cdif 3063   u. cun 3064   <.cop 3525   class class class wbr 3924   X. cxp 4532   `'ccnv 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542

This theorem is referenced by: (None)