Theorem swoord1 6410

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 3166	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
5	3, 4	eqsstri 3093	. . . . . 6 |- R C_ ( X X. X )
6	5	ssbri 3935	. . . . 5 |- ( A R B -> A ( X X. X ) B )
7		df-br 3894	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
8		opelxp1 4531	. . . . . 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 4185	. . . 4 |- ( ( ph /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( A .< C -> ( A .< B \/ B .< C ) ) )
15	1, 10, 11, 12, 14	syl13anc 1199	. . 3 |- ( ph -> ( A .< C -> ( A .< B \/ B .< C ) ) )
16	3	brdifun 6408	. . . . . . 7 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
17	10, 12, 16	syl2anc 406	. . . . . 6 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	2, 17	mpbid 146	. . . . 5 |- ( ph -> -. ( A .< B \/ B .< A ) )
19		orc 684	. . . . 5 |- ( A .< B -> ( A .< B \/ B .< A ) )
20	18, 19	nsyl 600	. . . 4 |- ( ph -> -. A .< B )
21		biorf 716	. . . 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 4185	. . . 4 |- ( ( ph /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( B .< C -> ( B .< A \/ A .< C ) ) )
25	1, 12, 11, 10, 24	syl13anc 1199	. . 3 |- ( ph -> ( B .< C -> ( B .< A \/ A .< C ) ) )
26		olc 683	. . . . 5 |- ( B .< A -> ( A .< B \/ B .< A ) )
27	18, 26	nsyl 600	. . . 4 |- ( ph -> -. B .< A )
28		biorf 716	. . . 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 680   /\ w3a 943   = wceq 1312   e. wcel 1461   \ cdif 3032   u. cun 3033   <.cop 3494   class class class wbr 3893   X. cxp 4495   `'ccnv 4496

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505

This theorem is referenced by: (None)