Step |
Hyp |
Ref |
Expression |
1 |
|
xpord2.1 |
|- T = { <. x , y >. | ( x e. ( A X. B ) /\ y e. ( A X. B ) /\ ( ( ( 1st ` x ) R ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ ( ( 2nd ` x ) S ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ x =/= y ) ) } |
2 |
|
poxp2.1 |
|- ( ph -> R Po A ) |
3 |
|
poxp2.2 |
|- ( ph -> S Po B ) |
4 |
|
elxp2 |
|- ( a e. ( A X. B ) <-> E. p e. A E. q e. B a = <. p , q >. ) |
5 |
|
equid |
|- p = p |
6 |
|
equid |
|- q = q |
7 |
5 6
|
pm3.2i |
|- ( p = p /\ q = q ) |
8 |
|
neorian |
|- ( ( p =/= p \/ q =/= q ) <-> -. ( p = p /\ q = q ) ) |
9 |
8
|
con2bii |
|- ( ( p = p /\ q = q ) <-> -. ( p =/= p \/ q =/= q ) ) |
10 |
7 9
|
mpbi |
|- -. ( p =/= p \/ q =/= q ) |
11 |
|
simp3 |
|- ( ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) -> ( p =/= p \/ q =/= q ) ) |
12 |
10 11
|
mto |
|- -. ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) |
13 |
|
simp3 |
|- ( ( ( p e. A /\ q e. B ) /\ ( p e. A /\ q e. B ) /\ ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) ) -> ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) ) |
14 |
12 13
|
mto |
|- -. ( ( p e. A /\ q e. B ) /\ ( p e. A /\ q e. B ) /\ ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) ) |
15 |
1
|
xpord2lem |
|- ( <. p , q >. T <. p , q >. <-> ( ( p e. A /\ q e. B ) /\ ( p e. A /\ q e. B ) /\ ( ( p R p \/ p = p ) /\ ( q S q \/ q = q ) /\ ( p =/= p \/ q =/= q ) ) ) ) |
16 |
14 15
|
mtbir |
|- -. <. p , q >. T <. p , q >. |
17 |
|
breq12 |
|- ( ( a = <. p , q >. /\ a = <. p , q >. ) -> ( a T a <-> <. p , q >. T <. p , q >. ) ) |
18 |
17
|
anidms |
|- ( a = <. p , q >. -> ( a T a <-> <. p , q >. T <. p , q >. ) ) |
19 |
16 18
|
mtbiri |
|- ( a = <. p , q >. -> -. a T a ) |
20 |
19
|
rexlimivw |
|- ( E. q e. B a = <. p , q >. -> -. a T a ) |
21 |
20
|
rexlimivw |
|- ( E. p e. A E. q e. B a = <. p , q >. -> -. a T a ) |
22 |
4 21
|
sylbi |
|- ( a e. ( A X. B ) -> -. a T a ) |
23 |
22
|
adantl |
|- ( ( ph /\ a e. ( A X. B ) ) -> -. a T a ) |
24 |
|
3reeanv |
|- ( E. p e. A E. r e. A E. t e. A ( E. q e. B a = <. p , q >. /\ E. s e. B b = <. r , s >. /\ E. u e. B c = <. t , u >. ) <-> ( E. p e. A E. q e. B a = <. p , q >. /\ E. r e. A E. s e. B b = <. r , s >. /\ E. t e. A E. u e. B c = <. t , u >. ) ) |
25 |
|
3reeanv |
|- ( E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) <-> ( E. q e. B a = <. p , q >. /\ E. s e. B b = <. r , s >. /\ E. u e. B c = <. t , u >. ) ) |
26 |
25
|
rexbii |
|- ( E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) <-> E. t e. A ( E. q e. B a = <. p , q >. /\ E. s e. B b = <. r , s >. /\ E. u e. B c = <. t , u >. ) ) |
27 |
26
|
2rexbii |
|- ( E. p e. A E. r e. A E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) <-> E. p e. A E. r e. A E. t e. A ( E. q e. B a = <. p , q >. /\ E. s e. B b = <. r , s >. /\ E. u e. B c = <. t , u >. ) ) |
28 |
|
elxp2 |
|- ( b e. ( A X. B ) <-> E. r e. A E. s e. B b = <. r , s >. ) |
29 |
|
elxp2 |
|- ( c e. ( A X. B ) <-> E. t e. A E. u e. B c = <. t , u >. ) |
30 |
4 28 29
|
3anbi123i |
|- ( ( a e. ( A X. B ) /\ b e. ( A X. B ) /\ c e. ( A X. B ) ) <-> ( E. p e. A E. q e. B a = <. p , q >. /\ E. r e. A E. s e. B b = <. r , s >. /\ E. t e. A E. u e. B c = <. t , u >. ) ) |
31 |
24 27 30
|
3bitr4ri |
|- ( ( a e. ( A X. B ) /\ b e. ( A X. B ) /\ c e. ( A X. B ) ) <-> E. p e. A E. r e. A E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) ) |
32 |
|
df-3an |
|- ( ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) <-> ( ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) ) /\ ( s e. B /\ u e. B ) ) ) |
33 |
|
simp3 |
|- ( ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) -> ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) |
34 |
|
simp3 |
|- ( ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) -> ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) |
35 |
|
simpr1l |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> p e. A ) |
36 |
35
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> p e. A ) |
37 |
|
simpr2r |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> q e. B ) |
38 |
37
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> q e. B ) |
39 |
36 38
|
jca |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p e. A /\ q e. B ) ) |
40 |
|
simpr2l |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> t e. A ) |
41 |
40
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> t e. A ) |
42 |
|
simpr3r |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> u e. B ) |
43 |
42
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> u e. B ) |
44 |
41 43
|
jca |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( t e. A /\ u e. B ) ) |
45 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> R Po A ) |
46 |
|
simpr1r |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> r e. A ) |
47 |
46
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> r e. A ) |
48 |
|
potr |
|- ( ( R Po A /\ ( p e. A /\ r e. A /\ t e. A ) ) -> ( ( p R r /\ r R t ) -> p R t ) ) |
49 |
45 36 47 41 48
|
syl13anc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p R r /\ r R t ) -> p R t ) ) |
50 |
|
orc |
|- ( p R t -> ( p R t \/ p = t ) ) |
51 |
49 50
|
syl6 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p R r /\ r R t ) -> ( p R t \/ p = t ) ) ) |
52 |
51
|
expd |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p R r -> ( r R t -> ( p R t \/ p = t ) ) ) ) |
53 |
|
breq1 |
|- ( p = r -> ( p R t <-> r R t ) ) |
54 |
53 50
|
syl6bir |
|- ( p = r -> ( r R t -> ( p R t \/ p = t ) ) ) |
55 |
54
|
a1i |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p = r -> ( r R t -> ( p R t \/ p = t ) ) ) ) |
56 |
|
simprl1 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p R r \/ p = r ) ) |
57 |
52 55 56
|
mpjaod |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r R t -> ( p R t \/ p = t ) ) ) |
58 |
|
breq2 |
|- ( r = t -> ( p R r <-> p R t ) ) |
59 |
|
equequ2 |
|- ( r = t -> ( p = r <-> p = t ) ) |
60 |
58 59
|
orbi12d |
|- ( r = t -> ( ( p R r \/ p = r ) <-> ( p R t \/ p = t ) ) ) |
61 |
56 60
|
syl5ibcom |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r = t -> ( p R t \/ p = t ) ) ) |
62 |
|
simprr1 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r R t \/ r = t ) ) |
63 |
57 61 62
|
mpjaod |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p R t \/ p = t ) ) |
64 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> S Po B ) |
65 |
|
simpr3l |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> s e. B ) |
66 |
65
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> s e. B ) |
67 |
|
potr |
|- ( ( S Po B /\ ( q e. B /\ s e. B /\ u e. B ) ) -> ( ( q S s /\ s S u ) -> q S u ) ) |
68 |
64 38 66 43 67
|
syl13anc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( q S s /\ s S u ) -> q S u ) ) |
69 |
|
orc |
|- ( q S u -> ( q S u \/ q = u ) ) |
70 |
68 69
|
syl6 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( q S s /\ s S u ) -> ( q S u \/ q = u ) ) ) |
71 |
70
|
expd |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( q S s -> ( s S u -> ( q S u \/ q = u ) ) ) ) |
72 |
|
breq1 |
|- ( q = s -> ( q S u <-> s S u ) ) |
73 |
72 69
|
syl6bir |
|- ( q = s -> ( s S u -> ( q S u \/ q = u ) ) ) |
74 |
73
|
a1i |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( q = s -> ( s S u -> ( q S u \/ q = u ) ) ) ) |
75 |
|
simprl2 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( q S s \/ q = s ) ) |
76 |
71 74 75
|
mpjaod |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( s S u -> ( q S u \/ q = u ) ) ) |
77 |
|
breq2 |
|- ( s = u -> ( q S s <-> q S u ) ) |
78 |
|
equequ2 |
|- ( s = u -> ( q = s <-> q = u ) ) |
79 |
77 78
|
orbi12d |
|- ( s = u -> ( ( q S s \/ q = s ) <-> ( q S u \/ q = u ) ) ) |
80 |
75 79
|
syl5ibcom |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( s = u -> ( q S u \/ q = u ) ) ) |
81 |
|
simprr2 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( s S u \/ s = u ) ) |
82 |
76 80 81
|
mpjaod |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( q S u \/ q = u ) ) |
83 |
|
simprr3 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r =/= t \/ s =/= u ) ) |
84 |
|
neorian |
|- ( ( r =/= t \/ s =/= u ) <-> -. ( r = t /\ s = u ) ) |
85 |
83 84
|
sylib |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> -. ( r = t /\ s = u ) ) |
86 |
|
neorian |
|- ( ( p =/= t \/ q =/= u ) <-> -. ( p = t /\ q = u ) ) |
87 |
86
|
con2bii |
|- ( ( p = t /\ q = u ) <-> -. ( p =/= t \/ q =/= u ) ) |
88 |
|
breq1 |
|- ( p = t -> ( p R r <-> t R r ) ) |
89 |
|
equequ1 |
|- ( p = t -> ( p = r <-> t = r ) ) |
90 |
88 89
|
orbi12d |
|- ( p = t -> ( ( p R r \/ p = r ) <-> ( t R r \/ t = r ) ) ) |
91 |
90
|
adantr |
|- ( ( p = t /\ q = u ) -> ( ( p R r \/ p = r ) <-> ( t R r \/ t = r ) ) ) |
92 |
|
breq1 |
|- ( q = u -> ( q S s <-> u S s ) ) |
93 |
|
equequ1 |
|- ( q = u -> ( q = s <-> u = s ) ) |
94 |
92 93
|
orbi12d |
|- ( q = u -> ( ( q S s \/ q = s ) <-> ( u S s \/ u = s ) ) ) |
95 |
94
|
adantl |
|- ( ( p = t /\ q = u ) -> ( ( q S s \/ q = s ) <-> ( u S s \/ u = s ) ) ) |
96 |
|
neeq1 |
|- ( p = t -> ( p =/= r <-> t =/= r ) ) |
97 |
96
|
adantr |
|- ( ( p = t /\ q = u ) -> ( p =/= r <-> t =/= r ) ) |
98 |
|
neeq1 |
|- ( q = u -> ( q =/= s <-> u =/= s ) ) |
99 |
98
|
adantl |
|- ( ( p = t /\ q = u ) -> ( q =/= s <-> u =/= s ) ) |
100 |
97 99
|
orbi12d |
|- ( ( p = t /\ q = u ) -> ( ( p =/= r \/ q =/= s ) <-> ( t =/= r \/ u =/= s ) ) ) |
101 |
91 95 100
|
3anbi123d |
|- ( ( p = t /\ q = u ) -> ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) <-> ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) ) ) |
102 |
101
|
anbi1d |
|- ( ( p = t /\ q = u ) -> ( ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) <-> ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) ) |
103 |
102
|
anbi2d |
|- ( ( p = t /\ q = u ) -> ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) <-> ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) ) ) |
104 |
|
simprl1 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( t R r \/ t = r ) ) |
105 |
|
simprr1 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r R t \/ r = t ) ) |
106 |
|
orcom |
|- ( ( ( t R r /\ r R t ) \/ r = t ) <-> ( r = t \/ ( t R r /\ r R t ) ) ) |
107 |
|
ordi |
|- ( ( r = t \/ ( t R r /\ r R t ) ) <-> ( ( r = t \/ t R r ) /\ ( r = t \/ r R t ) ) ) |
108 |
|
orcom |
|- ( ( r = t \/ t R r ) <-> ( t R r \/ r = t ) ) |
109 |
|
equcom |
|- ( r = t <-> t = r ) |
110 |
109
|
orbi2i |
|- ( ( t R r \/ r = t ) <-> ( t R r \/ t = r ) ) |
111 |
108 110
|
bitri |
|- ( ( r = t \/ t R r ) <-> ( t R r \/ t = r ) ) |
112 |
|
orcom |
|- ( ( r = t \/ r R t ) <-> ( r R t \/ r = t ) ) |
113 |
111 112
|
anbi12i |
|- ( ( ( r = t \/ t R r ) /\ ( r = t \/ r R t ) ) <-> ( ( t R r \/ t = r ) /\ ( r R t \/ r = t ) ) ) |
114 |
106 107 113
|
3bitri |
|- ( ( ( t R r /\ r R t ) \/ r = t ) <-> ( ( t R r \/ t = r ) /\ ( r R t \/ r = t ) ) ) |
115 |
104 105 114
|
sylanbrc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( t R r /\ r R t ) \/ r = t ) ) |
116 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> R Po A ) |
117 |
40
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> t e. A ) |
118 |
46
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> r e. A ) |
119 |
|
po2nr |
|- ( ( R Po A /\ ( t e. A /\ r e. A ) ) -> -. ( t R r /\ r R t ) ) |
120 |
116 117 118 119
|
syl12anc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> -. ( t R r /\ r R t ) ) |
121 |
115 120
|
orcnd |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> r = t ) |
122 |
|
simprl2 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( u S s \/ u = s ) ) |
123 |
|
simprr2 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( s S u \/ s = u ) ) |
124 |
|
orcom |
|- ( ( ( u S s /\ s S u ) \/ s = u ) <-> ( s = u \/ ( u S s /\ s S u ) ) ) |
125 |
|
ordi |
|- ( ( s = u \/ ( u S s /\ s S u ) ) <-> ( ( s = u \/ u S s ) /\ ( s = u \/ s S u ) ) ) |
126 |
|
orcom |
|- ( ( s = u \/ u S s ) <-> ( u S s \/ s = u ) ) |
127 |
|
equcom |
|- ( s = u <-> u = s ) |
128 |
127
|
orbi2i |
|- ( ( u S s \/ s = u ) <-> ( u S s \/ u = s ) ) |
129 |
126 128
|
bitri |
|- ( ( s = u \/ u S s ) <-> ( u S s \/ u = s ) ) |
130 |
|
orcom |
|- ( ( s = u \/ s S u ) <-> ( s S u \/ s = u ) ) |
131 |
129 130
|
anbi12i |
|- ( ( ( s = u \/ u S s ) /\ ( s = u \/ s S u ) ) <-> ( ( u S s \/ u = s ) /\ ( s S u \/ s = u ) ) ) |
132 |
124 125 131
|
3bitri |
|- ( ( ( u S s /\ s S u ) \/ s = u ) <-> ( ( u S s \/ u = s ) /\ ( s S u \/ s = u ) ) ) |
133 |
122 123 132
|
sylanbrc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( u S s /\ s S u ) \/ s = u ) ) |
134 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> S Po B ) |
135 |
42
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> u e. B ) |
136 |
65
|
adantr |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> s e. B ) |
137 |
|
po2nr |
|- ( ( S Po B /\ ( u e. B /\ s e. B ) ) -> -. ( u S s /\ s S u ) ) |
138 |
134 135 136 137
|
syl12anc |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> -. ( u S s /\ s S u ) ) |
139 |
133 138
|
orcnd |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> s = u ) |
140 |
121 139
|
jca |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( t R r \/ t = r ) /\ ( u S s \/ u = s ) /\ ( t =/= r \/ u =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r = t /\ s = u ) ) |
141 |
103 140
|
syl6bi |
|- ( ( p = t /\ q = u ) -> ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( r = t /\ s = u ) ) ) |
142 |
141
|
com12 |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p = t /\ q = u ) -> ( r = t /\ s = u ) ) ) |
143 |
87 142
|
syl5bir |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( -. ( p =/= t \/ q =/= u ) -> ( r = t /\ s = u ) ) ) |
144 |
85 143
|
mt3d |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( p =/= t \/ q =/= u ) ) |
145 |
63 82 144
|
3jca |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) |
146 |
39 44 145
|
3jca |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) /\ ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) |
147 |
146
|
ex |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> ( ( ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) -> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) ) |
148 |
33 34 147
|
syl2ani |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> ( ( ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) /\ ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) ) |
149 |
|
breq12 |
|- ( ( a = <. p , q >. /\ b = <. r , s >. ) -> ( a T b <-> <. p , q >. T <. r , s >. ) ) |
150 |
149
|
3adant3 |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( a T b <-> <. p , q >. T <. r , s >. ) ) |
151 |
1
|
xpord2lem |
|- ( <. p , q >. T <. r , s >. <-> ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) ) |
152 |
150 151
|
bitrdi |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( a T b <-> ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) ) ) |
153 |
|
breq12 |
|- ( ( b = <. r , s >. /\ c = <. t , u >. ) -> ( b T c <-> <. r , s >. T <. t , u >. ) ) |
154 |
153
|
3adant1 |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( b T c <-> <. r , s >. T <. t , u >. ) ) |
155 |
1
|
xpord2lem |
|- ( <. r , s >. T <. t , u >. <-> ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) |
156 |
154 155
|
bitrdi |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( b T c <-> ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) ) |
157 |
152 156
|
anbi12d |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) <-> ( ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) /\ ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) ) ) |
158 |
|
breq12 |
|- ( ( a = <. p , q >. /\ c = <. t , u >. ) -> ( a T c <-> <. p , q >. T <. t , u >. ) ) |
159 |
158
|
3adant2 |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( a T c <-> <. p , q >. T <. t , u >. ) ) |
160 |
1
|
xpord2lem |
|- ( <. p , q >. T <. t , u >. <-> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) |
161 |
159 160
|
bitrdi |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( a T c <-> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) ) |
162 |
157 161
|
imbi12d |
|- ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( ( a T b /\ b T c ) -> a T c ) <-> ( ( ( ( p e. A /\ q e. B ) /\ ( r e. A /\ s e. B ) /\ ( ( p R r \/ p = r ) /\ ( q S s \/ q = s ) /\ ( p =/= r \/ q =/= s ) ) ) /\ ( ( r e. A /\ s e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( r R t \/ r = t ) /\ ( s S u \/ s = u ) /\ ( r =/= t \/ s =/= u ) ) ) ) -> ( ( p e. A /\ q e. B ) /\ ( t e. A /\ u e. B ) /\ ( ( p R t \/ p = t ) /\ ( q S u \/ q = u ) /\ ( p =/= t \/ q =/= u ) ) ) ) ) ) |
163 |
148 162
|
syl5ibrcom |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) /\ ( s e. B /\ u e. B ) ) ) -> ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
164 |
32 163
|
sylan2br |
|- ( ( ph /\ ( ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) ) /\ ( s e. B /\ u e. B ) ) ) -> ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
165 |
164
|
anassrs |
|- ( ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) ) ) /\ ( s e. B /\ u e. B ) ) -> ( ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
166 |
165
|
rexlimdvva |
|- ( ( ph /\ ( ( p e. A /\ r e. A ) /\ ( t e. A /\ q e. B ) ) ) -> ( E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
167 |
166
|
anassrs |
|- ( ( ( ph /\ ( p e. A /\ r e. A ) ) /\ ( t e. A /\ q e. B ) ) -> ( E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
168 |
167
|
rexlimdvva |
|- ( ( ph /\ ( p e. A /\ r e. A ) ) -> ( E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
169 |
168
|
rexlimdvva |
|- ( ph -> ( E. p e. A E. r e. A E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) -> ( ( a T b /\ b T c ) -> a T c ) ) ) |
170 |
169
|
imp |
|- ( ( ph /\ E. p e. A E. r e. A E. t e. A E. q e. B E. s e. B E. u e. B ( a = <. p , q >. /\ b = <. r , s >. /\ c = <. t , u >. ) ) -> ( ( a T b /\ b T c ) -> a T c ) ) |
171 |
31 170
|
sylan2b |
|- ( ( ph /\ ( a e. ( A X. B ) /\ b e. ( A X. B ) /\ c e. ( A X. B ) ) ) -> ( ( a T b /\ b T c ) -> a T c ) ) |
172 |
23 171
|
ispod |
|- ( ph -> T Po ( A X. B ) ) |