Theorem swoord1 7340

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 22	. . . 4 |- ( ph -> ph )
2		swoord.6	. . . . 5 |- ( ph -> A R B )
3		swoer.1	. . . . . . 7 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
4		difss 3631	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
5	3, 4	eqsstri 3534	. . . . . 6 |- R C_ ( X X. X )
6	5	ssbri 4489	. . . . 5 |- ( A R B -> A ( X X. X ) B )
7		df-br 4448	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
8		opelxp1 5032	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
9	7, 8	sylbi 195	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
10	2, 6, 9	3syl 20	. . . 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 4809	. . . 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 7338	. . . . . . 7 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
17	10, 12, 16	syl2anc 661	. . . . . 6 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	2, 17	mpbid 210	. . . . 5 |- ( ph -> -. ( A .< B \/ B .< A ) )
19		orc 385	. . . . 5 |- ( A .< B -> ( A .< B \/ B .< A ) )
20	18, 19	nsyl 121	. . . 4 |- ( ph -> -. A .< B )
21		biorf 405	. . . 4 |- ( -. A .< B -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
22	20, 21	syl 16	. . 3 |- ( ph -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
23	15, 22	sylibrd 234	. 2 |- ( ph -> ( A .< C -> B .< C ) )
24	13	swopolem 4809	. . . 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 384	. . . . 5 |- ( B .< A -> ( A .< B \/ B .< A ) )
27	18, 26	nsyl 121	. . . 4 |- ( ph -> -. B .< A )
28		biorf 405	. . . 4 |- ( -. B .< A -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
29	27, 28	syl 16	. . 3 |- ( ph -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
30	25, 29	sylibrd 234	. 2 |- ( ph -> ( B .< C -> A .< C ) )
31	23, 30	impbid 191	1 |- ( ph -> ( A .< C <-> B .< C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   \ cdif 3473   u. cun 3474   <.cop 4033   class class class wbr 4447   X. cxp 4997   `'ccnv 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007

This theorem is referenced by: (None)