Theorem swoord2 6195

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 19	. . . 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 3099	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
6	4, 5	eqsstri 3030	. . . . . 6 |- R C_ ( X X. X )
7	6	ssbri 3829	. . . . 5 |- ( A R B -> A ( X X. X ) B )
8		df-br 3788	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
9		opelxp1 4397	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
10	8, 9	sylbi 119	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
11	3, 7, 10	3syl 17	. . . 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 4062	. . . 4 |- ( ( ph /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( C .< A -> ( C .< B \/ B .< A ) ) )
15	1, 2, 11, 12, 14	syl13anc 1172	. . 3 |- ( ph -> ( C .< A -> ( C .< B \/ B .< A ) ) )
16		idd 21	. . . 4 |- ( ph -> ( C .< B -> C .< B ) )
17	4	brdifun 6192	. . . . . . . 8 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	11, 12, 17	syl2anc 403	. . . . . . 7 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
19	3, 18	mpbid 145	. . . . . 6 |- ( ph -> -. ( A .< B \/ B .< A ) )
20		olc 665	. . . . . 6 |- ( B .< A -> ( A .< B \/ B .< A ) )
21	19, 20	nsyl 591	. . . . 5 |- ( ph -> -. B .< A )
22	21	pm2.21d 582	. . . 4 |- ( ph -> ( B .< A -> C .< B ) )
23	16, 22	jaod 670	. . 3 |- ( ph -> ( ( C .< B \/ B .< A ) -> C .< B ) )
24	15, 23	syld 44	. 2 |- ( ph -> ( C .< A -> C .< B ) )
25	13	swopolem 4062	. . . 4 |- ( ( ph /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C .< B -> ( C .< A \/ A .< B ) ) )
26	1, 2, 12, 11, 25	syl13anc 1172	. . 3 |- ( ph -> ( C .< B -> ( C .< A \/ A .< B ) ) )
27		idd 21	. . . 4 |- ( ph -> ( C .< A -> C .< A ) )
28		orc 666	. . . . . 6 |- ( A .< B -> ( A .< B \/ B .< A ) )
29	19, 28	nsyl 591	. . . . 5 |- ( ph -> -. A .< B )
30	29	pm2.21d 582	. . . 4 |- ( ph -> ( A .< B -> C .< A ) )
31	27, 30	jaod 670	. . 3 |- ( ph -> ( ( C .< A \/ A .< B ) -> C .< A ) )
32	26, 31	syld 44	. 2 |- ( ph -> ( C .< B -> C .< A ) )
33	24, 32	impbid 127	1 |- ( ph -> ( C .< A <-> C .< B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662   /\ w3a 920   = wceq 1285   e. wcel 1434   \ cdif 2971   u. cun 2972   <.cop 3403   class class class wbr 3787   X. cxp 4363   `'ccnv 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373

This theorem is referenced by: (None)