Step |
Hyp |
Ref |
Expression |
1 |
|
relowlpssretop.1 |
|
2 |
1
|
relowlssretop |
|
3 |
|
2re |
|
4 |
|
1lt2 |
|
5 |
|
ovex |
|
6 |
|
sbcan |
|
7 |
|
1re |
|
8 |
|
sbcg |
|
9 |
7 8
|
ax-mp |
|
10 |
|
sbcbr123 |
|
11 |
|
csbvarg |
|
12 |
7 11
|
ax-mp |
|
13 |
|
csbconstg |
|
14 |
7 13
|
ax-mp |
|
15 |
12 14
|
breq12i |
|
16 |
|
csbconstg |
|
17 |
7 16
|
ax-mp |
|
18 |
17
|
breqi |
|
19 |
10 15 18
|
3bitri |
|
20 |
9 19
|
anbi12i |
|
21 |
6 20
|
bitri |
|
22 |
|
sbceqg |
|
23 |
7 22
|
ax-mp |
|
24 |
|
csbconstg |
|
25 |
7 24
|
ax-mp |
|
26 |
|
csbov123 |
|
27 |
|
csbconstg |
|
28 |
7 27
|
ax-mp |
|
29 |
12 14 28
|
oveq123i |
|
30 |
26 29
|
eqtri |
|
31 |
25 30
|
eqeq12i |
|
32 |
23 31
|
bitri |
|
33 |
|
sbcan |
|
34 |
|
simpr |
|
35 |
|
simpl |
|
36 |
|
leid |
|
37 |
35 36
|
jccir |
|
38 |
|
rexr |
|
39 |
|
elico2 |
|
40 |
38 39
|
sylan2 |
|
41 |
|
df-3an |
|
42 |
40 41
|
syl6bb |
|
43 |
42
|
baibd |
|
44 |
37 43
|
mpdan |
|
45 |
44
|
biimpar |
|
46 |
45
|
adantr |
|
47 |
|
eleq2 |
|
48 |
47
|
adantl |
|
49 |
46 48
|
mpbird |
|
50 |
|
rexpssxrxp |
|
51 |
|
opelxpi |
|
52 |
50 51
|
sseldi |
|
53 |
|
df-ico |
|
54 |
53
|
ixxf |
|
55 |
54
|
fdmi |
|
56 |
55
|
eleq2i |
|
57 |
53
|
mpofun |
|
58 |
|
funfvima |
|
59 |
57 58
|
mpan |
|
60 |
56 59
|
sylbir |
|
61 |
52 51 60
|
sylc |
|
62 |
|
df-ov |
|
63 |
61 62 1
|
3eltr4g |
|
64 |
|
eleq1 |
|
65 |
63 64
|
syl5ibrcom |
|
66 |
65
|
imp |
|
67 |
|
ioof |
|
68 |
|
ffn |
|
69 |
67 68
|
ax-mp |
|
70 |
|
ovelrn |
|
71 |
69 70
|
ax-mp |
|
72 |
|
iooelexlt |
|
73 |
|
df-rex |
|
74 |
72 73
|
sylib |
|
75 |
|
simpl |
|
76 |
75
|
a1i |
|
77 |
53
|
elmpocl2 |
|
78 |
|
elioore |
|
79 |
|
elico2 |
|
80 |
78 79
|
sylan |
|
81 |
|
simp2 |
|
82 |
80 81
|
syl6bi |
|
83 |
82
|
ex |
|
84 |
83
|
com23 |
|
85 |
77 84
|
mpdi |
|
86 |
85
|
imp |
|
87 |
78
|
rexrd |
|
88 |
87
|
adantr |
|
89 |
|
elicore |
|
90 |
78 89
|
sylan |
|
91 |
90
|
rexrd |
|
92 |
|
xrlenlt |
|
93 |
92
|
biimpd |
|
94 |
93
|
con2d |
|
95 |
88 91 94
|
syl2anc |
|
96 |
86 95
|
mt2d |
|
97 |
96
|
intnand |
|
98 |
97
|
ex |
|
99 |
98
|
con2d |
|
100 |
76 99
|
jcad |
|
101 |
|
annim |
|
102 |
100 101
|
syl6ib |
|
103 |
102
|
eximdv |
|
104 |
74 103
|
mpd |
|
105 |
|
exnal |
|
106 |
104 105
|
sylib |
|
107 |
|
dfss2 |
|
108 |
106 107
|
sylnibr |
|
109 |
|
imnan |
|
110 |
108 109
|
mpbi |
|
111 |
|
eleq2 |
|
112 |
|
sseq1 |
|
113 |
111 112
|
anbi12d |
|
114 |
110 113
|
mtbiri |
|
115 |
|
sseq2 |
|
116 |
115
|
anbi2d |
|
117 |
116
|
notbid |
|
118 |
114 117
|
syl5ibrcom |
|
119 |
118
|
a1i |
|
120 |
119
|
rexlimivv |
|
121 |
71 120
|
sylbi |
|
122 |
121
|
com12 |
|
123 |
122
|
ralrimiv |
|
124 |
|
ralnex |
|
125 |
123 124
|
sylib |
|
126 |
125
|
adantl |
|
127 |
66 126
|
jca |
|
128 |
127
|
adantlr |
|
129 |
49 128
|
jca |
|
130 |
|
an12 |
|
131 |
|
annim |
|
132 |
131
|
anbi2i |
|
133 |
130 132
|
bitri |
|
134 |
129 133
|
sylib |
|
135 |
|
rspe |
|
136 |
134 135
|
syl |
|
137 |
|
rexnal |
|
138 |
136 137
|
sylib |
|
139 |
138
|
exp41 |
|
140 |
139
|
com4l |
|
141 |
140
|
imp41 |
|
142 |
|
rspe |
|
143 |
34 141 142
|
syl2anc |
|
144 |
|
rexnal |
|
145 |
143 144
|
sylib |
|
146 |
|
df-ico |
|
147 |
146
|
ixxex |
|
148 |
|
imaexg |
|
149 |
147 148
|
ax-mp |
|
150 |
1 149
|
eqeltri |
|
151 |
1
|
icoreunrn |
|
152 |
|
unirnioo |
|
153 |
151 152
|
eqtr3i |
|
154 |
|
tgss2 |
|
155 |
150 153 154
|
mp2an |
|
156 |
151
|
raleqi |
|
157 |
155 156
|
bitr4i |
|
158 |
145 157
|
sylnibr |
|
159 |
158
|
sbcth |
|
160 |
7 159
|
ax-mp |
|
161 |
|
sbcimg |
|
162 |
7 161
|
ax-mp |
|
163 |
160 162
|
mpbi |
|
164 |
|
sbcel1v |
|
165 |
7 164
|
mpbir |
|
166 |
|
sbcan |
|
167 |
165 166
|
mpbiran2 |
|
168 |
|
sbcg |
|
169 |
7 168
|
ax-mp |
|
170 |
163 167 169
|
3imtr3i |
|
171 |
33 170
|
sylbir |
|
172 |
21 32 171
|
syl2anbr |
|
173 |
172
|
sbcth |
|
174 |
5 173
|
ax-mp |
|
175 |
|
sbcimg |
|
176 |
5 175
|
ax-mp |
|
177 |
174 176
|
mpbi |
|
178 |
|
sbcan |
|
179 |
|
eqid |
|
180 |
|
eqsbc3 |
|
181 |
5 180
|
ax-mp |
|
182 |
179 181
|
mpbir |
|
183 |
|
sbcg |
|
184 |
5 183
|
ax-mp |
|
185 |
184
|
anbi1i |
|
186 |
182 185
|
mpbiran2 |
|
187 |
178 186
|
bitri |
|
188 |
|
sbcg |
|
189 |
5 188
|
ax-mp |
|
190 |
177 187 189
|
3imtr3i |
|
191 |
190
|
sbcth |
|
192 |
3 191
|
ax-mp |
|
193 |
|
sbcimg |
|
194 |
3 193
|
ax-mp |
|
195 |
192 194
|
mpbi |
|
196 |
|
sbcan |
|
197 |
|
sbcel1v |
|
198 |
|
sbcbr123 |
|
199 |
|
csbconstg |
|
200 |
3 199
|
ax-mp |
|
201 |
|
csbvarg |
|
202 |
3 201
|
ax-mp |
|
203 |
200 202
|
breq12i |
|
204 |
|
csbconstg |
|
205 |
3 204
|
ax-mp |
|
206 |
205
|
breqi |
|
207 |
198 203 206
|
3bitri |
|
208 |
197 207
|
anbi12i |
|
209 |
196 208
|
bitri |
|
210 |
|
sbcg |
|
211 |
3 210
|
ax-mp |
|
212 |
195 209 211
|
3imtr3i |
|
213 |
3 4 212
|
mp2an |
|
214 |
|
eqimss |
|
215 |
213 214
|
mto |
|
216 |
215
|
nesymir |
|
217 |
|
df-pss |
|
218 |
2 216 217
|
mpbir2an |
|