Theorem swoord2 6543

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 3253	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
6	4, 5	eqsstri 3179	. . . . . 6 |- R C_ ( X X. X )
7	6	ssbri 4033	. . . . 5 |- ( A R B -> A ( X X. X ) B )
8		df-br 3990	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
9		opelxp1 4645	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
10	8, 9	sylbi 120	. . . . 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 4290	. . . 4 |- ( ( ph /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( C .< A -> ( C .< B \/ B .< A ) ) )
15	1, 2, 11, 12, 14	syl13anc 1235	. . 3 |- ( ph -> ( C .< A -> ( C .< B \/ B .< A ) ) )
16		idd 21	. . . 4 |- ( ph -> ( C .< B -> C .< B ) )
17	4	brdifun 6540	. . . . . . . 8 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	11, 12, 17	syl2anc 409	. . . . . . 7 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
19	3, 18	mpbid 146	. . . . . 6 |- ( ph -> -. ( A .< B \/ B .< A ) )
20		olc 706	. . . . . 6 |- ( B .< A -> ( A .< B \/ B .< A ) )
21	19, 20	nsyl 623	. . . . 5 |- ( ph -> -. B .< A )
22	21	pm2.21d 614	. . . 4 |- ( ph -> ( B .< A -> C .< B ) )
23	16, 22	jaod 712	. . 3 |- ( ph -> ( ( C .< B \/ B .< A ) -> C .< B ) )
24	15, 23	syld 45	. 2 |- ( ph -> ( C .< A -> C .< B ) )
25	13	swopolem 4290	. . . 4 |- ( ( ph /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C .< B -> ( C .< A \/ A .< B ) ) )
26	1, 2, 12, 11, 25	syl13anc 1235	. . 3 |- ( ph -> ( C .< B -> ( C .< A \/ A .< B ) ) )
27		idd 21	. . . 4 |- ( ph -> ( C .< A -> C .< A ) )
28		orc 707	. . . . . 6 |- ( A .< B -> ( A .< B \/ B .< A ) )
29	19, 28	nsyl 623	. . . . 5 |- ( ph -> -. A .< B )
30	29	pm2.21d 614	. . . 4 |- ( ph -> ( A .< B -> C .< A ) )
31	27, 30	jaod 712	. . 3 |- ( ph -> ( ( C .< A \/ A .< B ) -> C .< A ) )
32	26, 31	syld 45	. 2 |- ( ph -> ( C .< B -> C .< A ) )
33	24, 32	impbid 128	1 |- ( ph -> ( C .< A <-> C .< B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   \ cdif 3118   u. cun 3119   <.cop 3586   class class class wbr 3989   X. cxp 4609   `'ccnv 4610

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619

This theorem is referenced by: (None)