Theorem swoord2 6938

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
swoord2	|- ( ph -> ( C .< A <-> C .< B ) )

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 swoord2

Step	Hyp	Ref	Expression
1		id 21	. . . 4 |- ( ph -> ph )
2		swoord.5	. . . 4 |- ( ph -> C e. X )
3		swoord.6	. . . . 5 |- ( ph -> A R B )
4		swoer.1	. . . . . . 7 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
5		difss 3476	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
6	4, 5	eqsstri 3380	. . . . . 6 |- R C_ ( X X. X )
7	6	ssbri 4257	. . . . 5 |- ( A R B -> A ( X X. X ) B )
8		df-br 4216	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
9		opelxp1 4914	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
10	8, 9	sylbi 189	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
11	3, 7, 10	3syl 19	. . . 4 |- ( ph -> A 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 4515	. . . 4 |- ( ( ph /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( C .< A -> ( C .< B \/ B .< A ) ) )
15	1, 2, 11, 12, 14	syl13anc 1187	. . 3 |- ( ph -> ( C .< A -> ( C .< B \/ B .< A ) ) )
16		idd 23	. . . 4 |- ( ph -> ( C .< B -> C .< B ) )
17	4	brdifun 6935	. . . . . . . 8 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	11, 12, 17	syl2anc 644	. . . . . . 7 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
19	3, 18	mpbid 203	. . . . . 6 |- ( ph -> -. ( A .< B \/ B .< A ) )
20		olc 375	. . . . . 6 |- ( B .< A -> ( A .< B \/ B .< A ) )
21	19, 20	nsyl 116	. . . . 5 |- ( ph -> -. B .< A )
22	21	pm2.21d 101	. . . 4 |- ( ph -> ( B .< A -> C .< B ) )
23	16, 22	jaod 371	. . 3 |- ( ph -> ( ( C .< B \/ B .< A ) -> C .< B ) )
24	15, 23	syld 43	. 2 |- ( ph -> ( C .< A -> C .< B ) )
25	13	swopolem 4515	. . . 4 |- ( ( ph /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C .< B -> ( C .< A \/ A .< B ) ) )
26	1, 2, 12, 11, 25	syl13anc 1187	. . 3 |- ( ph -> ( C .< B -> ( C .< A \/ A .< B ) ) )
27		idd 23	. . . 4 |- ( ph -> ( C .< A -> C .< A ) )
28		orc 376	. . . . . 6 |- ( A .< B -> ( A .< B \/ B .< A ) )
29	19, 28	nsyl 116	. . . . 5 |- ( ph -> -. A .< B )
30	29	pm2.21d 101	. . . 4 |- ( ph -> ( A .< B -> C .< A ) )
31	27, 30	jaod 371	. . 3 |- ( ph -> ( ( C .< A \/ A .< B ) -> C .< A ) )
32	26, 31	syld 43	. 2 |- ( ph -> ( C .< B -> C .< A ) )
33	24, 32	impbid 185	1 |- ( ph -> ( C .< A <-> C .< B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   \/ wo 359   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   \ cdif 3319   u. cun 3320   <.cop 3819   class class class wbr 4215   X. cxp 4879   `'ccnv 4880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889