Step |
Hyp |
Ref |
Expression |
1 |
|
scutcl |
|- ( A < ( A |s B ) e. No ) |
2 |
1
|
ad3antrrr |
|- ( ( ( ( A < ( A |s B ) e. No ) |
3 |
|
scutcl |
|- ( C < ( C |s D ) e. No ) |
4 |
3
|
ad3antlr |
|- ( ( ( ( A < ( C |s D ) e. No ) |
5 |
|
ssltss2 |
|- ( C < D C_ No ) |
6 |
5
|
ad2antlr |
|- ( ( ( A < D C_ No ) |
7 |
6
|
sselda |
|- ( ( ( ( A < d e. No ) |
8 |
|
simplr |
|- ( ( ( ( A < ( A |s B ) <_s ( C |s D ) ) |
9 |
|
scutcut |
|- ( C < ( ( C |s D ) e. No /\ C < |
10 |
9
|
simp3d |
|- ( C < { ( C |s D ) } < |
11 |
10
|
ad2antlr |
|- ( ( ( A < { ( C |s D ) } < |
12 |
|
ssltsep |
|- ( { ( C |s D ) } < A. a e. { ( C |s D ) } A. d e. D a |
13 |
11 12
|
syl |
|- ( ( ( A < A. a e. { ( C |s D ) } A. d e. D a |
14 |
|
ovex |
|- ( C |s D ) e. _V |
15 |
|
breq1 |
|- ( a = ( C |s D ) -> ( a ( C |s D ) |
16 |
15
|
ralbidv |
|- ( a = ( C |s D ) -> ( A. d e. D a A. d e. D ( C |s D ) |
17 |
14 16
|
ralsn |
|- ( A. a e. { ( C |s D ) } A. d e. D a A. d e. D ( C |s D ) |
18 |
13 17
|
sylib |
|- ( ( ( A < A. d e. D ( C |s D ) |
19 |
18
|
r19.21bi |
|- ( ( ( ( A < ( C |s D ) |
20 |
2 4 7 8 19
|
slelttrd |
|- ( ( ( ( A < ( A |s B ) |
21 |
20
|
ralrimiva |
|- ( ( ( A < A. d e. D ( A |s B ) |
22 |
|
ssltss1 |
|- ( A < A C_ No ) |
23 |
22
|
adantr |
|- ( ( A < A C_ No ) |
24 |
23
|
adantr |
|- ( ( ( A < A C_ No ) |
25 |
24
|
sselda |
|- ( ( ( ( A < a e. No ) |
26 |
1
|
ad3antrrr |
|- ( ( ( ( A < ( A |s B ) e. No ) |
27 |
3
|
ad3antlr |
|- ( ( ( ( A < ( C |s D ) e. No ) |
28 |
|
scutcut |
|- ( A < ( ( A |s B ) e. No /\ A < |
29 |
28
|
simp2d |
|- ( A < A < |
30 |
29
|
adantr |
|- ( ( A < A < |
31 |
30
|
adantr |
|- ( ( ( A < A < |
32 |
|
ssltsep |
|- ( A < A. a e. A A. d e. { ( A |s B ) } a |
33 |
31 32
|
syl |
|- ( ( ( A < A. a e. A A. d e. { ( A |s B ) } a |
34 |
33
|
r19.21bi |
|- ( ( ( ( A < A. d e. { ( A |s B ) } a |
35 |
|
ovex |
|- ( A |s B ) e. _V |
36 |
|
breq2 |
|- ( d = ( A |s B ) -> ( a a |
37 |
35 36
|
ralsn |
|- ( A. d e. { ( A |s B ) } a a |
38 |
34 37
|
sylib |
|- ( ( ( ( A < a |
39 |
|
simplr |
|- ( ( ( ( A < ( A |s B ) <_s ( C |s D ) ) |
40 |
25 26 27 38 39
|
sltletrd |
|- ( ( ( ( A < a |
41 |
40
|
ralrimiva |
|- ( ( ( A < A. a e. A a |
42 |
21 41
|
jca |
|- ( ( ( A < ( A. d e. D ( A |s B ) |
43 |
|
bdayelon |
|- ( bday ` ( A |s B ) ) e. On |
44 |
43
|
onordi |
|- Ord ( bday ` ( A |s B ) ) |
45 |
|
ordn2lp |
|- ( Ord ( bday ` ( A |s B ) ) -> -. ( ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) /\ ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) ) |
46 |
44 45
|
ax-mp |
|- -. ( ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) /\ ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) |
47 |
3
|
ad2antlr |
|- ( ( ( A < ( C |s D ) e. No ) |
48 |
1
|
adantr |
|- ( ( A < ( A |s B ) e. No ) |
49 |
48
|
adantr |
|- ( ( ( A < ( A |s B ) e. No ) |
50 |
|
sltnle |
|- ( ( ( C |s D ) e. No /\ ( A |s B ) e. No ) -> ( ( C |s D ) -. ( A |s B ) <_s ( C |s D ) ) ) |
51 |
47 49 50
|
syl2anc |
|- ( ( ( A < ( ( C |s D ) -. ( A |s B ) <_s ( C |s D ) ) ) |
52 |
3
|
ad3antlr |
|- ( ( ( ( A < ( C |s D ) e. No ) |
53 |
|
ssltex1 |
|- ( A < A e. _V ) |
54 |
53
|
ad3antrrr |
|- ( ( ( ( A < A e. _V ) |
55 |
|
snex |
|- { ( C |s D ) } e. _V |
56 |
54 55
|
jctir |
|- ( ( ( ( A < ( A e. _V /\ { ( C |s D ) } e. _V ) ) |
57 |
22
|
ad3antrrr |
|- ( ( ( ( A < A C_ No ) |
58 |
52
|
snssd |
|- ( ( ( ( A < { ( C |s D ) } C_ No ) |
59 |
|
simplrr |
|- ( ( ( ( A < A. a e. A a |
60 |
|
breq2 |
|- ( d = ( C |s D ) -> ( a a |
61 |
14 60
|
ralsn |
|- ( A. d e. { ( C |s D ) } a a |
62 |
61
|
ralbii |
|- ( A. a e. A A. d e. { ( C |s D ) } a A. a e. A a |
63 |
59 62
|
sylibr |
|- ( ( ( ( A < A. a e. A A. d e. { ( C |s D ) } a |
64 |
57 58 63
|
3jca |
|- ( ( ( ( A < ( A C_ No /\ { ( C |s D ) } C_ No /\ A. a e. A A. d e. { ( C |s D ) } a |
65 |
|
brsslt |
|- ( A < ( ( A e. _V /\ { ( C |s D ) } e. _V ) /\ ( A C_ No /\ { ( C |s D ) } C_ No /\ A. a e. A A. d e. { ( C |s D ) } a |
66 |
56 64 65
|
sylanbrc |
|- ( ( ( ( A < A < |
67 |
|
ssltex2 |
|- ( A < B e. _V ) |
68 |
67
|
ad3antrrr |
|- ( ( ( ( A < B e. _V ) |
69 |
68 55
|
jctil |
|- ( ( ( ( A < ( { ( C |s D ) } e. _V /\ B e. _V ) ) |
70 |
|
ssltss2 |
|- ( A < B C_ No ) |
71 |
70
|
ad3antrrr |
|- ( ( ( ( A < B C_ No ) |
72 |
52
|
adantr |
|- ( ( ( ( ( A < ( C |s D ) e. No ) |
73 |
48
|
ad3antrrr |
|- ( ( ( ( ( A < ( A |s B ) e. No ) |
74 |
71
|
sselda |
|- ( ( ( ( ( A < b e. No ) |
75 |
|
simplr |
|- ( ( ( ( ( A < ( C |s D ) |
76 |
28
|
simp3d |
|- ( A < { ( A |s B ) } < |
77 |
76
|
ad3antrrr |
|- ( ( ( ( A < { ( A |s B ) } < |
78 |
|
ssltsep |
|- ( { ( A |s B ) } < A. a e. { ( A |s B ) } A. b e. B a |
79 |
77 78
|
syl |
|- ( ( ( ( A < A. a e. { ( A |s B ) } A. b e. B a |
80 |
|
breq1 |
|- ( a = ( A |s B ) -> ( a ( A |s B ) |
81 |
80
|
ralbidv |
|- ( a = ( A |s B ) -> ( A. b e. B a A. b e. B ( A |s B ) |
82 |
35 81
|
ralsn |
|- ( A. a e. { ( A |s B ) } A. b e. B a A. b e. B ( A |s B ) |
83 |
79 82
|
sylib |
|- ( ( ( ( A < A. b e. B ( A |s B ) |
84 |
83
|
r19.21bi |
|- ( ( ( ( ( A < ( A |s B ) |
85 |
72 73 74 75 84
|
slttrd |
|- ( ( ( ( ( A < ( C |s D ) |
86 |
85
|
ralrimiva |
|- ( ( ( ( A < A. b e. B ( C |s D ) |
87 |
|
breq1 |
|- ( a = ( C |s D ) -> ( a ( C |s D ) |
88 |
87
|
ralbidv |
|- ( a = ( C |s D ) -> ( A. b e. B a A. b e. B ( C |s D ) |
89 |
14 88
|
ralsn |
|- ( A. a e. { ( C |s D ) } A. b e. B a A. b e. B ( C |s D ) |
90 |
86 89
|
sylibr |
|- ( ( ( ( A < A. a e. { ( C |s D ) } A. b e. B a |
91 |
58 71 90
|
3jca |
|- ( ( ( ( A < ( { ( C |s D ) } C_ No /\ B C_ No /\ A. a e. { ( C |s D ) } A. b e. B a |
92 |
|
brsslt |
|- ( { ( C |s D ) } < ( ( { ( C |s D ) } e. _V /\ B e. _V ) /\ ( { ( C |s D ) } C_ No /\ B C_ No /\ A. a e. { ( C |s D ) } A. b e. B a |
93 |
69 91 92
|
sylanbrc |
|- ( ( ( ( A < { ( C |s D ) } < |
94 |
|
sltirr |
|- ( ( A |s B ) e. No -> -. ( A |s B ) |
95 |
49 94
|
syl |
|- ( ( ( A < -. ( A |s B ) |
96 |
|
breq1 |
|- ( ( A |s B ) = ( C |s D ) -> ( ( A |s B ) ( C |s D ) |
97 |
96
|
notbid |
|- ( ( A |s B ) = ( C |s D ) -> ( -. ( A |s B ) -. ( C |s D ) |
98 |
95 97
|
syl5ibcom |
|- ( ( ( A < ( ( A |s B ) = ( C |s D ) -> -. ( C |s D ) |
99 |
98
|
necon2ad |
|- ( ( ( A < ( ( C |s D ) ( A |s B ) =/= ( C |s D ) ) ) |
100 |
99
|
imp |
|- ( ( ( ( A < ( A |s B ) =/= ( C |s D ) ) |
101 |
100
|
necomd |
|- ( ( ( ( A < ( C |s D ) =/= ( A |s B ) ) |
102 |
|
scutbdaylt |
|- ( ( ( C |s D ) e. No /\ ( A < ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) ) |
103 |
52 66 93 101 102
|
syl121anc |
|- ( ( ( ( A < ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) ) |
104 |
1
|
ad3antrrr |
|- ( ( ( ( A < ( A |s B ) e. No ) |
105 |
|
ssltex1 |
|- ( C < C e. _V ) |
106 |
105
|
ad3antlr |
|- ( ( ( ( A < C e. _V ) |
107 |
|
snex |
|- { ( A |s B ) } e. _V |
108 |
106 107
|
jctir |
|- ( ( ( ( A < ( C e. _V /\ { ( A |s B ) } e. _V ) ) |
109 |
|
ssltss1 |
|- ( C < C C_ No ) |
110 |
109
|
ad3antlr |
|- ( ( ( ( A < C C_ No ) |
111 |
104
|
snssd |
|- ( ( ( ( A < { ( A |s B ) } C_ No ) |
112 |
110
|
sselda |
|- ( ( ( ( ( A < c e. No ) |
113 |
52
|
adantr |
|- ( ( ( ( ( A < ( C |s D ) e. No ) |
114 |
48
|
ad3antrrr |
|- ( ( ( ( ( A < ( A |s B ) e. No ) |
115 |
9
|
simp2d |
|- ( C < C < |
116 |
115
|
ad3antlr |
|- ( ( ( ( A < C < |
117 |
|
ssltsep |
|- ( C < A. c e. C A. d e. { ( C |s D ) } c |
118 |
116 117
|
syl |
|- ( ( ( ( A < A. c e. C A. d e. { ( C |s D ) } c |
119 |
118
|
r19.21bi |
|- ( ( ( ( ( A < A. d e. { ( C |s D ) } c |
120 |
|
breq2 |
|- ( d = ( C |s D ) -> ( c c |
121 |
14 120
|
ralsn |
|- ( A. d e. { ( C |s D ) } c c |
122 |
119 121
|
sylib |
|- ( ( ( ( ( A < c |
123 |
|
simplr |
|- ( ( ( ( ( A < ( C |s D ) |
124 |
112 113 114 122 123
|
slttrd |
|- ( ( ( ( ( A < c |
125 |
|
breq2 |
|- ( a = ( A |s B ) -> ( c c |
126 |
35 125
|
ralsn |
|- ( A. a e. { ( A |s B ) } c c |
127 |
124 126
|
sylibr |
|- ( ( ( ( ( A < A. a e. { ( A |s B ) } c |
128 |
127
|
ralrimiva |
|- ( ( ( ( A < A. c e. C A. a e. { ( A |s B ) } c |
129 |
110 111 128
|
3jca |
|- ( ( ( ( A < ( C C_ No /\ { ( A |s B ) } C_ No /\ A. c e. C A. a e. { ( A |s B ) } c |
130 |
|
brsslt |
|- ( C < ( ( C e. _V /\ { ( A |s B ) } e. _V ) /\ ( C C_ No /\ { ( A |s B ) } C_ No /\ A. c e. C A. a e. { ( A |s B ) } c |
131 |
108 129 130
|
sylanbrc |
|- ( ( ( ( A < C < |
132 |
|
ssltex2 |
|- ( C < D e. _V ) |
133 |
132
|
ad3antlr |
|- ( ( ( ( A < D e. _V ) |
134 |
133 107
|
jctil |
|- ( ( ( ( A < ( { ( A |s B ) } e. _V /\ D e. _V ) ) |
135 |
5
|
ad3antlr |
|- ( ( ( ( A < D C_ No ) |
136 |
|
simplrl |
|- ( ( ( ( A < A. d e. D ( A |s B ) |
137 |
|
breq1 |
|- ( a = ( A |s B ) -> ( a ( A |s B ) |
138 |
137
|
ralbidv |
|- ( a = ( A |s B ) -> ( A. d e. D a A. d e. D ( A |s B ) |
139 |
35 138
|
ralsn |
|- ( A. a e. { ( A |s B ) } A. d e. D a A. d e. D ( A |s B ) |
140 |
136 139
|
sylibr |
|- ( ( ( ( A < A. a e. { ( A |s B ) } A. d e. D a |
141 |
111 135 140
|
3jca |
|- ( ( ( ( A < ( { ( A |s B ) } C_ No /\ D C_ No /\ A. a e. { ( A |s B ) } A. d e. D a |
142 |
|
brsslt |
|- ( { ( A |s B ) } < ( ( { ( A |s B ) } e. _V /\ D e. _V ) /\ ( { ( A |s B ) } C_ No /\ D C_ No /\ A. a e. { ( A |s B ) } A. d e. D a |
143 |
134 141 142
|
sylanbrc |
|- ( ( ( ( A < { ( A |s B ) } < |
144 |
|
scutbdaylt |
|- ( ( ( A |s B ) e. No /\ ( C < ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) |
145 |
104 131 143 100 144
|
syl121anc |
|- ( ( ( ( A < ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) |
146 |
103 145
|
jca |
|- ( ( ( ( A < ( ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) /\ ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) ) |
147 |
146
|
ex |
|- ( ( ( A < ( ( C |s D ) ( ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) /\ ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) ) ) |
148 |
51 147
|
sylbird |
|- ( ( ( A < ( -. ( A |s B ) <_s ( C |s D ) -> ( ( bday ` ( A |s B ) ) e. ( bday ` ( C |s D ) ) /\ ( bday ` ( C |s D ) ) e. ( bday ` ( A |s B ) ) ) ) ) |
149 |
46 148
|
mt3i |
|- ( ( ( A < ( A |s B ) <_s ( C |s D ) ) |
150 |
42 149
|
impbida |
|- ( ( A < ( ( A |s B ) <_s ( C |s D ) <-> ( A. d e. D ( A |s B ) |
151 |
|
breq12 |
|- ( ( X = ( A |s B ) /\ Y = ( C |s D ) ) -> ( X <_s Y <-> ( A |s B ) <_s ( C |s D ) ) ) |
152 |
|
breq1 |
|- ( X = ( A |s B ) -> ( X ( A |s B ) |
153 |
152
|
ralbidv |
|- ( X = ( A |s B ) -> ( A. d e. D X A. d e. D ( A |s B ) |
154 |
|
breq2 |
|- ( Y = ( C |s D ) -> ( a a |
155 |
154
|
ralbidv |
|- ( Y = ( C |s D ) -> ( A. a e. A a A. a e. A a |
156 |
153 155
|
bi2anan9 |
|- ( ( X = ( A |s B ) /\ Y = ( C |s D ) ) -> ( ( A. d e. D X ( A. d e. D ( A |s B ) |
157 |
151 156
|
bibi12d |
|- ( ( X = ( A |s B ) /\ Y = ( C |s D ) ) -> ( ( X <_s Y <-> ( A. d e. D X ( ( A |s B ) <_s ( C |s D ) <-> ( A. d e. D ( A |s B ) |
158 |
150 157
|
syl5ibr |
|- ( ( X = ( A |s B ) /\ Y = ( C |s D ) ) -> ( ( A < ( X <_s Y <-> ( A. d e. D X |
159 |
158
|
impcom |
|- ( ( ( A < ( X <_s Y <-> ( A. d e. D X |