Theorem swoer 6457

Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

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 ) ) )

Assertion

Ref	Expression
swoer	|- ( ph -> R Er X )

Distinct variable groups:   x, y, z, .<   ph, x, y, z   x, X, y, z

Allowed substitution hints:   R( x, y, z)

Proof of Theorem swoer

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		swoer.1	. . . . 5 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
2		difss 3202	. . . . 5 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
3	1, 2	eqsstri 3129	. . . 4 |- R C_ ( X X. X )
4		relxp 4648	. . . 4 |- Rel ( X X. X )
5		relss 4626	. . . 4 |- ( R C_ ( X X. X ) -> ( Rel ( X X. X ) -> Rel R ) )
6	3, 4, 5	mp2 16	. . 3 |- Rel R
7	6	a1i 9	. 2 |- ( ph -> Rel R )
8		simpr 109	. . 3 |- ( ( ph /\ u R v ) -> u R v )
9		orcom 717	. . . . . 6 |- ( ( u .< v \/ v .< u ) <-> ( v .< u \/ u .< v ) )
10	9	a1i 9	. . . . 5 |- ( ( ph /\ u R v ) -> ( ( u .< v \/ v .< u ) <-> ( v .< u \/ u .< v ) ) )
11	10	notbid 656	. . . 4 |- ( ( ph /\ u R v ) -> ( -. ( u .< v \/ v .< u ) <-> -. ( v .< u \/ u .< v ) ) )
12	3	ssbri 3972	. . . . . . 7 |- ( u R v -> u ( X X. X ) v )
13	12	adantl 275	. . . . . 6 |- ( ( ph /\ u R v ) -> u ( X X. X ) v )
14		brxp 4570	. . . . . 6 |- ( u ( X X. X ) v <-> ( u e. X /\ v e. X ) )
15	13, 14	sylib 121	. . . . 5 |- ( ( ph /\ u R v ) -> ( u e. X /\ v e. X ) )
16	1	brdifun 6456	. . . . 5 |- ( ( u e. X /\ v e. X ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
17	15, 16	syl 14	. . . 4 |- ( ( ph /\ u R v ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
18	15	simprd 113	. . . . 5 |- ( ( ph /\ u R v ) -> v e. X )
19	15	simpld 111	. . . . 5 |- ( ( ph /\ u R v ) -> u e. X )
20	1	brdifun 6456	. . . . 5 |- ( ( v e. X /\ u e. X ) -> ( v R u <-> -. ( v .< u \/ u .< v ) ) )
21	18, 19, 20	syl2anc 408	. . . 4 |- ( ( ph /\ u R v ) -> ( v R u <-> -. ( v .< u \/ u .< v ) ) )
22	11, 17, 21	3bitr4d 219	. . 3 |- ( ( ph /\ u R v ) -> ( u R v <-> v R u ) )
23	8, 22	mpbid 146	. 2 |- ( ( ph /\ u R v ) -> v R u )
24		simprl 520	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u R v )
25	12	ad2antrl 481	. . . . . . 7 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u ( X X. X ) v )
26	14	simplbi 272	. . . . . . 7 |- ( u ( X X. X ) v -> u e. X )
27	25, 26	syl 14	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u e. X )
28	14	simprbi 273	. . . . . . 7 |- ( u ( X X. X ) v -> v e. X )
29	25, 28	syl 14	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> v e. X )
30	27, 29, 16	syl2anc 408	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
31	24, 30	mpbid 146	. . . 4 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( u .< v \/ v .< u ) )
32		simprr 521	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> v R w )
33	3	brel 4591	. . . . . . . 8 |- ( v R w -> ( v e. X /\ w e. X ) )
34	33	simprd 113	. . . . . . 7 |- ( v R w -> w e. X )
35	32, 34	syl 14	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> w e. X )
36	1	brdifun 6456	. . . . . 6 |- ( ( v e. X /\ w e. X ) -> ( v R w <-> -. ( v .< w \/ w .< v ) ) )
37	29, 35, 36	syl2anc 408	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( v R w <-> -. ( v .< w \/ w .< v ) ) )
38	32, 37	mpbid 146	. . . 4 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( v .< w \/ w .< v ) )
39		simpl 108	. . . . . . 7 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ph )
40		swoer.3	. . . . . . . 8 |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )
41	40	swopolem 4227	. . . . . . 7 |- ( ( ph /\ ( u e. X /\ w e. X /\ v e. X ) ) -> ( u .< w -> ( u .< v \/ v .< w ) ) )
42	39, 27, 35, 29, 41	syl13anc 1218	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u .< w -> ( u .< v \/ v .< w ) ) )
43	40	swopolem 4227	. . . . . . . 8 |- ( ( ph /\ ( w e. X /\ u e. X /\ v e. X ) ) -> ( w .< u -> ( w .< v \/ v .< u ) ) )
44	39, 35, 27, 29, 43	syl13anc 1218	. . . . . . 7 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( w .< u -> ( w .< v \/ v .< u ) ) )
45		orcom 717	. . . . . . 7 |- ( ( v .< u \/ w .< v ) <-> ( w .< v \/ v .< u ) )
46	44, 45	syl6ibr 161	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( w .< u -> ( v .< u \/ w .< v ) ) )
47	42, 46	orim12d 775	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( ( u .< w \/ w .< u ) -> ( ( u .< v \/ v .< w ) \/ ( v .< u \/ w .< v ) ) ) )
48		or4 760	. . . . 5 |- ( ( ( u .< v \/ v .< w ) \/ ( v .< u \/ w .< v ) ) <-> ( ( u .< v \/ v .< u ) \/ ( v .< w \/ w .< v ) ) )
49	47, 48	syl6ib 160	. . . 4 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( ( u .< w \/ w .< u ) -> ( ( u .< v \/ v .< u ) \/ ( v .< w \/ w .< v ) ) ) )
50	31, 38, 49	mtord 772	. . 3 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( u .< w \/ w .< u ) )
51	1	brdifun 6456	. . . 4 |- ( ( u e. X /\ w e. X ) -> ( u R w <-> -. ( u .< w \/ w .< u ) ) )
52	27, 35, 51	syl2anc 408	. . 3 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u R w <-> -. ( u .< w \/ w .< u ) ) )
53	50, 52	mpbird 166	. 2 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u R w )
54		swoer.2	. . . . . . 7 |- ( ( ph /\ ( y e. X /\ z e. X ) ) -> ( y .< z -> -. z .< y ) )
55	54, 40	swopo 4228	. . . . . 6 |- ( ph -> .< Po X )
56		poirr 4229	. . . . . 6 |- ( ( .< Po X /\ u e. X ) -> -. u .< u )
57	55, 56	sylan 281	. . . . 5 |- ( ( ph /\ u e. X ) -> -. u .< u )
58		pm1.2 745	. . . . 5 |- ( ( u .< u \/ u .< u ) -> u .< u )
59	57, 58	nsyl 617	. . . 4 |- ( ( ph /\ u e. X ) -> -. ( u .< u \/ u .< u ) )
60		simpr 109	. . . . 5 |- ( ( ph /\ u e. X ) -> u e. X )
61	1	brdifun 6456	. . . . 5 |- ( ( u e. X /\ u e. X ) -> ( u R u <-> -. ( u .< u \/ u .< u ) ) )
62	60, 60, 61	syl2anc 408	. . . 4 |- ( ( ph /\ u e. X ) -> ( u R u <-> -. ( u .< u \/ u .< u ) ) )
63	59, 62	mpbird 166	. . 3 |- ( ( ph /\ u e. X ) -> u R u )
64	3	ssbri 3972	. . . . 5 |- ( u R u -> u ( X X. X ) u )
65		brxp 4570	. . . . . 6 |- ( u ( X X. X ) u <-> ( u e. X /\ u e. X ) )
66	65	simplbi 272	. . . . 5 |- ( u ( X X. X ) u -> u e. X )
67	64, 66	syl 14	. . . 4 |- ( u R u -> u e. X )
68	67	adantl 275	. . 3 |- ( ( ph /\ u R u ) -> u e. X )
69	63, 68	impbida 585	. 2 |- ( ph -> ( u e. X <-> u R u ) )
70	7, 23, 53, 69	iserd 6455	1 |- ( ph -> R Er X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   \ cdif 3068   u. cun 3069   C_ wss 3071   class class class wbr 3929   Po wpo 4216   X. cxp 4537   `'ccnv 4538   Rel wrel 4544   Er wer 6426

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-po 4218  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-er 6429

This theorem is referenced by: (None)