Theorem swoord1 7410

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 3549	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
5	3, 4	eqsstri 3448	. . . . . 6 |- R C_ ( X X. X )
6	5	ssbri 4438	. . . . 5 |- ( A R B -> A ( X X. X ) B )
7		df-br 4396	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
8		opelxp1 4872	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
9	7, 8	sylbi 200	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
10	2, 6, 9	3syl 18	. . . 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 4769	. . . 4 |- ( ( ph /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( A .< C -> ( A .< B \/ B .< C ) ) )
15	1, 10, 11, 12, 14	syl13anc 1294	. . 3 |- ( ph -> ( A .< C -> ( A .< B \/ B .< C ) ) )
16	3	brdifun 7408	. . . . . . 7 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
17	10, 12, 16	syl2anc 673	. . . . . 6 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	2, 17	mpbid 215	. . . . 5 |- ( ph -> -. ( A .< B \/ B .< A ) )
19		orc 392	. . . . 5 |- ( A .< B -> ( A .< B \/ B .< A ) )
20	18, 19	nsyl 125	. . . 4 |- ( ph -> -. A .< B )
21		biorf 412	. . . 4 |- ( -. A .< B -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
22	20, 21	syl 17	. . 3 |- ( ph -> ( B .< C <-> ( A .< B \/ B .< C ) ) )
23	15, 22	sylibrd 242	. 2 |- ( ph -> ( A .< C -> B .< C ) )
24	13	swopolem 4769	. . . 4 |- ( ( ph /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( B .< C -> ( B .< A \/ A .< C ) ) )
25	1, 12, 11, 10, 24	syl13anc 1294	. . 3 |- ( ph -> ( B .< C -> ( B .< A \/ A .< C ) ) )
26		olc 391	. . . . 5 |- ( B .< A -> ( A .< B \/ B .< A ) )
27	18, 26	nsyl 125	. . . 4 |- ( ph -> -. B .< A )
28		biorf 412	. . . 4 |- ( -. B .< A -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
29	27, 28	syl 17	. . 3 |- ( ph -> ( A .< C <-> ( B .< A \/ A .< C ) ) )
30	25, 29	sylibrd 242	. 2 |- ( ph -> ( B .< C -> A .< C ) )
31	23, 30	impbid 195	1 |- ( ph -> ( A .< C <-> B .< C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   \ cdif 3387   u. cun 3388   <.cop 3965   class class class wbr 4395   X. cxp 4837   `'ccnv 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847

This theorem is referenced by: (None)