Theorem swoord1 6530

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 3248	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
5	3, 4	eqsstri 3174	. . . . . 6 |- R C_ ( X X. X )
6	5	ssbri 4026	. . . . 5 |- ( A R B -> A ( X X. X ) B )
7		df-br 3983	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
8		opelxp1 4638	. . . . . 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 4283	. . . 4 |- ( ( ph /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( A .< C -> ( A .< B \/ B .< C ) ) )
15	1, 10, 11, 12, 14	syl13anc 1230	. . 3 |- ( ph -> ( A .< C -> ( A .< B \/ B .< C ) ) )
16	3	brdifun 6528	. . . . . . 7 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
17	10, 12, 16	syl2anc 409	. . . . . 6 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	2, 17	mpbid 146	. . . . 5 |- ( ph -> -. ( A .< B \/ B .< A ) )
19		orc 702	. . . . 5 |- ( A .< B -> ( A .< B \/ B .< A ) )
20	18, 19	nsyl 618	. . . 4 |- ( ph -> -. A .< B )
21		biorf 734	. . . 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 4283	. . . 4 |- ( ( ph /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( B .< C -> ( B .< A \/ A .< C ) ) )
25	1, 12, 11, 10, 24	syl13anc 1230	. . 3 |- ( ph -> ( B .< C -> ( B .< A \/ A .< C ) ) )
26		olc 701	. . . . 5 |- ( B .< A -> ( A .< B \/ B .< A ) )
27	18, 26	nsyl 618	. . . 4 |- ( ph -> -. B .< A )
28		biorf 734	. . . 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 698   /\ w3a 968   = wceq 1343   e. wcel 2136   \ cdif 3113   u. cun 3114   <.cop 3579   class class class wbr 3982   X. cxp 4602   `'ccnv 4603

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612

This theorem is referenced by: (None)