Theorem swoord2 6894

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 20	. . . 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 3434	. . . . . . 7 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
6	4, 5	eqsstri 3338	. . . . . 6 |- R C_ ( X X. X )
7	6	ssbri 4214	. . . . 5 |- ( A R B -> A ( X X. X ) B )
8		df-br 4173	. . . . . 6 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
9		opelxp1 4870	. . . . . 6 |- ( <. A , B >. e. ( X X. X ) -> A e. X )
10	8, 9	sylbi 188	. . . . 5 |- ( A ( X X. X ) B -> A e. X )
11	3, 7, 10	3syl 19	. . . 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 4472	. . . 4 |- ( ( ph /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( C .< A -> ( C .< B \/ B .< A ) ) )
15	1, 2, 11, 12, 14	syl13anc 1186	. . 3 |- ( ph -> ( C .< A -> ( C .< B \/ B .< A ) ) )
16		idd 22	. . . 4 |- ( ph -> ( C .< B -> C .< B ) )
17	4	brdifun 6891	. . . . . . . 8 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
18	11, 12, 17	syl2anc 643	. . . . . . 7 |- ( ph -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
19	3, 18	mpbid 202	. . . . . 6 |- ( ph -> -. ( A .< B \/ B .< A ) )
20		olc 374	. . . . . 6 |- ( B .< A -> ( A .< B \/ B .< A ) )
21	19, 20	nsyl 115	. . . . 5 |- ( ph -> -. B .< A )
22	21	pm2.21d 100	. . . 4 |- ( ph -> ( B .< A -> C .< B ) )
23	16, 22	jaod 370	. . 3 |- ( ph -> ( ( C .< B \/ B .< A ) -> C .< B ) )
24	15, 23	syld 42	. 2 |- ( ph -> ( C .< A -> C .< B ) )
25	13	swopolem 4472	. . . 4 |- ( ( ph /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C .< B -> ( C .< A \/ A .< B ) ) )
26	1, 2, 12, 11, 25	syl13anc 1186	. . 3 |- ( ph -> ( C .< B -> ( C .< A \/ A .< B ) ) )
27		idd 22	. . . 4 |- ( ph -> ( C .< A -> C .< A ) )
28		orc 375	. . . . . 6 |- ( A .< B -> ( A .< B \/ B .< A ) )
29	19, 28	nsyl 115	. . . . 5 |- ( ph -> -. A .< B )
30	29	pm2.21d 100	. . . 4 |- ( ph -> ( A .< B -> C .< A ) )
31	27, 30	jaod 370	. . 3 |- ( ph -> ( ( C .< A \/ A .< B ) -> C .< A ) )
32	26, 31	syld 42	. 2 |- ( ph -> ( C .< B -> C .< A ) )
33	24, 32	impbid 184	1 |- ( ph -> ( C .< A <-> C .< B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   \ cdif 3277   u. cun 3278   <.cop 3777   class class class wbr 4172   X. cxp 4835   `'ccnv 4836

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845