Theorem swoord2 6567

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 3263	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
6	4, 5	eqsstri 3189	. . . . . 6 |- R C_ ( X X. X )
7	6	ssbri 4049	. . . . 5 |- ( A R B -> A ( X X. X ) B )
8		df-br 4006	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
9		opelxp1 4662	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
10	8, 9	sylbi 121	. . . . 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 4307	. . . 4 |- ( ( ph /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( C .< A -> ( C .< B \/ B .< A ) ) )
15	1, 2, 11, 12, 14	syl13anc 1240	. . 3 |- ( ph -> ( C .< A -> ( C .< B \/ B .< A ) ) )
16		idd 21	. . . 4 |- ( ph -> ( C .< B -> C .< B ) )
17	4	brdifun 6564	. . . . . . . 8 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	11, 12, 17	syl2anc 411	. . . . . . 7 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
19	3, 18	mpbid 147	. . . . . 6 |- ( ph -> -. ( A .< B \/ B .< A ) )
20		olc 711	. . . . . 6 |- ( B .< A -> ( A .< B \/ B .< A ) )
21	19, 20	nsyl 628	. . . . 5 |- ( ph -> -. B .< A )
22	21	pm2.21d 619	. . . 4 |- ( ph -> ( B .< A -> C .< B ) )
23	16, 22	jaod 717	. . 3 |- ( ph -> ( ( C .< B \/ B .< A ) -> C .< B ) )
24	15, 23	syld 45	. 2 |- ( ph -> ( C .< A -> C .< B ) )
25	13	swopolem 4307	. . . 4 |- ( ( ph /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C .< B -> ( C .< A \/ A .< B ) ) )
26	1, 2, 12, 11, 25	syl13anc 1240	. . 3 |- ( ph -> ( C .< B -> ( C .< A \/ A .< B ) ) )
27		idd 21	. . . 4 |- ( ph -> ( C .< A -> C .< A ) )
28		orc 712	. . . . . 6 |- ( A .< B -> ( A .< B \/ B .< A ) )
29	19, 28	nsyl 628	. . . . 5 |- ( ph -> -. A .< B )
30	29	pm2.21d 619	. . . 4 |- ( ph -> ( A .< B -> C .< A ) )
31	27, 30	jaod 717	. . 3 |- ( ph -> ( ( C .< A \/ A .< B ) -> C .< A ) )
32	26, 31	syld 45	. 2 |- ( ph -> ( C .< B -> C .< A ) )
33	24, 32	impbid 129	1 |- ( ph -> ( C .< A <-> C .< B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   \ cdif 3128   u. cun 3129   <.cop 3597   class class class wbr 4005   X. cxp 4626   `'ccnv 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636

This theorem is referenced by: (None)