Theorem swoer 7772

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 3737	. . . . 5 |- ( ( X X. X ) \ ( .< u. `' .< ) ) C_ ( X X. X )
3	1, 2	eqsstri 3635	. . . 4 |- R C_ ( X X. X )
4		relxp 5227	. . . 4 |- Rel ( X X. X )
5		relss 5206	. . . 4 |- ( R C_ ( X X. X ) -> ( Rel ( X X. X ) -> Rel R ) )
6	3, 4, 5	mp2 9	. . 3 |- Rel R
7	6	a1i 11	. 2 |- ( ph -> Rel R )
8		simpr 477	. . 3 |- ( ( ph /\ u R v ) -> u R v )
9		orcom 402	. . . . . 6 |- ( ( u .< v \/ v .< u ) <-> ( v .< u \/ u .< v ) )
10	9	a1i 11	. . . . 5 |- ( ( ph /\ u R v ) -> ( ( u .< v \/ v .< u ) <-> ( v .< u \/ u .< v ) ) )
11	10	notbid 308	. . . 4 |- ( ( ph /\ u R v ) -> ( -. ( u .< v \/ v .< u ) <-> -. ( v .< u \/ u .< v ) ) )
12	3	ssbri 4697	. . . . . . 7 |- ( u R v -> u ( X X. X ) v )
13	12	adantl 482	. . . . . 6 |- ( ( ph /\ u R v ) -> u ( X X. X ) v )
14		brxp 5147	. . . . . 6 |- ( u ( X X. X ) v <-> ( u e. X /\ v e. X ) )
15	13, 14	sylib 208	. . . . 5 |- ( ( ph /\ u R v ) -> ( u e. X /\ v e. X ) )
16	1	brdifun 7771	. . . . 5 |- ( ( u e. X /\ v e. X ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
17	15, 16	syl 17	. . . 4 |- ( ( ph /\ u R v ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
18	15	simprd 479	. . . . 5 |- ( ( ph /\ u R v ) -> v e. X )
19	15	simpld 475	. . . . 5 |- ( ( ph /\ u R v ) -> u e. X )
20	1	brdifun 7771	. . . . 5 |- ( ( v e. X /\ u e. X ) -> ( v R u <-> -. ( v .< u \/ u .< v ) ) )
21	18, 19, 20	syl2anc 693	. . . 4 |- ( ( ph /\ u R v ) -> ( v R u <-> -. ( v .< u \/ u .< v ) ) )
22	11, 17, 21	3bitr4d 300	. . 3 |- ( ( ph /\ u R v ) -> ( u R v <-> v R u ) )
23	8, 22	mpbid 222	. 2 |- ( ( ph /\ u R v ) -> v R u )
24		simprl 794	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u R v )
25	12	ad2antrl 764	. . . . . . 7 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u ( X X. X ) v )
26	14	simplbi 476	. . . . . . 7 |- ( u ( X X. X ) v -> u e. X )
27	25, 26	syl 17	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> u e. X )
28	14	simprbi 480	. . . . . . 7 |- ( u ( X X. X ) v -> v e. X )
29	25, 28	syl 17	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> v e. X )
30	27, 29, 16	syl2anc 693	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u R v <-> -. ( u .< v \/ v .< u ) ) )
31	24, 30	mpbid 222	. . . 4 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( u .< v \/ v .< u ) )
32		simprr 796	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> v R w )
33	3	brel 5168	. . . . . . . 8 |- ( v R w -> ( v e. X /\ w e. X ) )
34	33	simprd 479	. . . . . . 7 |- ( v R w -> w e. X )
35	32, 34	syl 17	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> w e. X )
36	1	brdifun 7771	. . . . . 6 |- ( ( v e. X /\ w e. X ) -> ( v R w <-> -. ( v .< w \/ w .< v ) ) )
37	29, 35, 36	syl2anc 693	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( v R w <-> -. ( v .< w \/ w .< v ) ) )
38	32, 37	mpbid 222	. . . 4 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( v .< w \/ w .< v ) )
39		simpl 473	. . . . . . 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 5044	. . . . . . 7 |- ( ( ph /\ ( u e. X /\ w e. X /\ v e. X ) ) -> ( u .< w -> ( u .< v \/ v .< w ) ) )
42	39, 27, 35, 29, 41	syl13anc 1328	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u .< w -> ( u .< v \/ v .< w ) ) )
43	40	swopolem 5044	. . . . . . . 8 |- ( ( ph /\ ( w e. X /\ u e. X /\ v e. X ) ) -> ( w .< u -> ( w .< v \/ v .< u ) ) )
44	39, 35, 27, 29, 43	syl13anc 1328	. . . . . . 7 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( w .< u -> ( w .< v \/ v .< u ) ) )
45		orcom 402	. . . . . . 7 |- ( ( v .< u \/ w .< v ) <-> ( w .< v \/ v .< u ) )
46	44, 45	syl6ibr 242	. . . . . 6 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( w .< u -> ( v .< u \/ w .< v ) ) )
47	42, 46	orim12d 883	. . . . 5 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( ( u .< w \/ w .< u ) -> ( ( u .< v \/ v .< w ) \/ ( v .< u \/ w .< v ) ) ) )
48		or4 550	. . . . 5 |- ( ( ( u .< v \/ v .< w ) \/ ( v .< u \/ w .< v ) ) <-> ( ( u .< v \/ v .< u ) \/ ( v .< w \/ w .< v ) ) )
49	47, 48	syl6ib 241	. . . 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 692	. . 3 |- ( ( ph /\ ( u R v /\ v R w ) ) -> -. ( u .< w \/ w .< u ) )
51	1	brdifun 7771	. . . 4 |- ( ( u e. X /\ w e. X ) -> ( u R w <-> -. ( u .< w \/ w .< u ) ) )
52	27, 35, 51	syl2anc 693	. . 3 |- ( ( ph /\ ( u R v /\ v R w ) ) -> ( u R w <-> -. ( u .< w \/ w .< u ) ) )
53	50, 52	mpbird 247	. 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 5045	. . . . . 6 |- ( ph -> .< Po X )
56		poirr 5046	. . . . . 6 |- ( ( .< Po X /\ u e. X ) -> -. u .< u )
57	55, 56	sylan 488	. . . . 5 |- ( ( ph /\ u e. X ) -> -. u .< u )
58		pm1.2 535	. . . . 5 |- ( ( u .< u \/ u .< u ) -> u .< u )
59	57, 58	nsyl 135	. . . 4 |- ( ( ph /\ u e. X ) -> -. ( u .< u \/ u .< u ) )
60		simpr 477	. . . . 5 |- ( ( ph /\ u e. X ) -> u e. X )
61	1	brdifun 7771	. . . . 5 |- ( ( u e. X /\ u e. X ) -> ( u R u <-> -. ( u .< u \/ u .< u ) ) )
62	60, 60, 61	syl2anc 693	. . . 4 |- ( ( ph /\ u e. X ) -> ( u R u <-> -. ( u .< u \/ u .< u ) ) )
63	59, 62	mpbird 247	. . 3 |- ( ( ph /\ u e. X ) -> u R u )
64	3	ssbri 4697	. . . . 5 |- ( u R u -> u ( X X. X ) u )
65		brxp 5147	. . . . . 6 |- ( u ( X X. X ) u <-> ( u e. X /\ u e. X ) )
66	65	simplbi 476	. . . . 5 |- ( u ( X X. X ) u -> u e. X )
67	64, 66	syl 17	. . . 4 |- ( u R u -> u e. X )
68	67	adantl 482	. . 3 |- ( ( ph /\ u R u ) -> u e. X )
69	63, 68	impbida 877	. 2 |- ( ph -> ( u e. X <-> u R u ) )
70	7, 23, 53, 69	iserd 7768	1 |- ( ph -> R Er X )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   C_ wss 3574   class class class wbr 4653   Po wpo 5033   X. cxp 5112   `'ccnv 5113   Rel wrel 5119   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by: (None)