Step |
Hyp |
Ref |
Expression |
1 |
|
canthp1lem2.1 |
|
2 |
|
canthp1lem2.2 |
|
3 |
|
canthp1lem2.3 |
|
4 |
|
canthp1lem2.4 |
|
5 |
|
canthp1lem2.5 |
|
6 |
|
canthp1lem2.6 |
|
7 |
|
relsdom |
|
8 |
7
|
brrelex2i |
|
9 |
1 8
|
syl |
|
10 |
9
|
pwexd |
|
11 |
|
f1oeng |
|
12 |
10 2 11
|
syl2anc |
|
13 |
12
|
ensymd |
|
14 |
|
canth2g |
|
15 |
9 14
|
syl |
|
16 |
|
sdomen2 |
|
17 |
12 16
|
syl |
|
18 |
15 17
|
mpbid |
|
19 |
|
sdomnen |
|
20 |
18 19
|
syl |
|
21 |
|
omelon |
|
22 |
|
onenon |
|
23 |
21 22
|
ax-mp |
|
24 |
|
dff1o3 |
|
25 |
24
|
simprbi |
|
26 |
2 25
|
syl |
|
27 |
|
f1ofo |
|
28 |
2 27
|
syl |
|
29 |
|
f1ofn |
|
30 |
|
fnresdm |
|
31 |
|
foeq1 |
|
32 |
2 29 30 31
|
4syl |
|
33 |
28 32
|
mpbird |
|
34 |
|
fvex |
|
35 |
|
f1osng |
|
36 |
9 34 35
|
sylancl |
|
37 |
2 29
|
syl |
|
38 |
|
pwidg |
|
39 |
9 38
|
syl |
|
40 |
|
fnressn |
|
41 |
37 39 40
|
syl2anc |
|
42 |
|
f1oeq1 |
|
43 |
41 42
|
syl |
|
44 |
36 43
|
mpbird |
|
45 |
|
f1ofo |
|
46 |
44 45
|
syl |
|
47 |
|
resdif |
|
48 |
26 33 46 47
|
syl3anc |
|
49 |
|
f1oco |
|
50 |
3 48 49
|
syl2anc |
|
51 |
|
resco |
|
52 |
|
f1oeq1 |
|
53 |
51 52
|
ax-mp |
|
54 |
50 53
|
sylibr |
|
55 |
|
f1of |
|
56 |
54 55
|
syl |
|
57 |
|
0elpw |
|
58 |
57
|
a1i |
|
59 |
|
sdom0 |
|
60 |
|
breq2 |
|
61 |
59 60
|
mtbii |
|
62 |
61
|
necon2ai |
|
63 |
1 62
|
syl |
|
64 |
63
|
ad2antrr |
|
65 |
|
eldifsn |
|
66 |
58 64 65
|
sylanbrc |
|
67 |
|
simplr |
|
68 |
|
simpr |
|
69 |
68
|
neqned |
|
70 |
|
eldifsn |
|
71 |
67 69 70
|
sylanbrc |
|
72 |
66 71
|
ifclda |
|
73 |
72
|
fmpttd |
|
74 |
56 73
|
fcod |
|
75 |
73
|
frnd |
|
76 |
|
cores |
|
77 |
75 76
|
syl |
|
78 |
77 4
|
eqtr4di |
|
79 |
78
|
feq1d |
|
80 |
74 79
|
mpbid |
|
81 |
|
inss1 |
|
82 |
81
|
a1i |
|
83 |
|
eqid |
|
84 |
5 6 83
|
canth4 |
|
85 |
9 80 82 84
|
syl3anc |
|
86 |
85
|
simp1d |
|
87 |
85
|
simp2d |
|
88 |
87
|
pssned |
|
89 |
88
|
necomd |
|
90 |
85
|
simp3d |
|
91 |
4
|
fveq1i |
|
92 |
4
|
fveq1i |
|
93 |
90 91 92
|
3eqtr3g |
|
94 |
9 86
|
sselpwd |
|
95 |
73 94
|
fvco3d |
|
96 |
87
|
pssssd |
|
97 |
96 86
|
sstrd |
|
98 |
9 97
|
sselpwd |
|
99 |
73 98
|
fvco3d |
|
100 |
93 95 99
|
3eqtr3d |
|
101 |
100
|
adantr |
|
102 |
|
eqid |
|
103 |
|
eqeq1 |
|
104 |
|
id |
|
105 |
103 104
|
ifbieq2d |
|
106 |
|
ifcl |
|
107 |
57 94 106
|
sylancr |
|
108 |
102 105 94 107
|
fvmptd3 |
|
109 |
|
pssne |
|
110 |
109
|
neneqd |
|
111 |
110
|
iffalsed |
|
112 |
108 111
|
sylan9eq |
|
113 |
112
|
fveq2d |
|
114 |
|
eqeq1 |
|
115 |
|
id |
|
116 |
114 115
|
ifbieq2d |
|
117 |
|
ifcl |
|
118 |
57 98 117
|
sylancr |
|
119 |
102 116 98 118
|
fvmptd3 |
|
120 |
119
|
adantr |
|
121 |
|
sspsstr |
|
122 |
96 121
|
sylan |
|
123 |
122
|
pssned |
|
124 |
123
|
neneqd |
|
125 |
124
|
iffalsed |
|
126 |
120 125
|
eqtrd |
|
127 |
126
|
fveq2d |
|
128 |
101 113 127
|
3eqtr3d |
|
129 |
94 109
|
anim12i |
|
130 |
|
eldifsn |
|
131 |
129 130
|
sylibr |
|
132 |
131
|
fvresd |
|
133 |
98
|
adantr |
|
134 |
|
eldifsn |
|
135 |
133 123 134
|
sylanbrc |
|
136 |
135
|
fvresd |
|
137 |
128 132 136
|
3eqtr4d |
|
138 |
|
f1of1 |
|
139 |
54 138
|
syl |
|
140 |
139
|
adantr |
|
141 |
|
f1fveq |
|
142 |
140 131 135 141
|
syl12anc |
|
143 |
137 142
|
mpbid |
|
144 |
143
|
ex |
|
145 |
144
|
necon3ad |
|
146 |
89 145
|
mpd |
|
147 |
|
npss |
|
148 |
146 147
|
sylib |
|
149 |
86 148
|
mpd |
|
150 |
|
eqid |
|
151 |
|
eqid |
|
152 |
150 151
|
pm3.2i |
|
153 |
|
elinel1 |
|
154 |
|
ffvelrn |
|
155 |
80 153 154
|
syl2an |
|
156 |
5 9 155 6
|
fpwwe |
|
157 |
152 156
|
mpbiri |
|
158 |
157
|
simpld |
|
159 |
5 9
|
fpwwelem |
|
160 |
158 159
|
mpbid |
|
161 |
160
|
simprld |
|
162 |
|
fvex |
|
163 |
|
weeq1 |
|
164 |
162 163
|
spcev |
|
165 |
161 164
|
syl |
|
166 |
|
ween |
|
167 |
165 166
|
sylibr |
|
168 |
149 167
|
eqeltrrd |
|
169 |
|
domtri2 |
|
170 |
23 168 169
|
sylancr |
|
171 |
|
infdju1 |
|
172 |
170 171
|
syl6bir |
|
173 |
|
ensym |
|
174 |
172 173
|
syl6 |
|
175 |
20 174
|
mt3d |
|
176 |
|
2onn |
|
177 |
|
nnsdom |
|
178 |
176 177
|
ax-mp |
|
179 |
|
djufi |
|
180 |
175 178 179
|
sylancl |
|
181 |
|
isfinite |
|
182 |
180 181
|
sylibr |
|
183 |
|
sssucid |
|
184 |
|
df-2o |
|
185 |
183 184
|
sseqtrri |
|
186 |
|
xpss2 |
|
187 |
185 186
|
ax-mp |
|
188 |
|
unss2 |
|
189 |
187 188
|
mp1i |
|
190 |
|
ssun2 |
|
191 |
|
1oex |
|
192 |
191
|
snid |
|
193 |
191
|
sucid |
|
194 |
193 184
|
eleqtrri |
|
195 |
|
opelxpi |
|
196 |
192 194 195
|
mp2an |
|
197 |
190 196
|
sselii |
|
198 |
|
1n0 |
|
199 |
198
|
neii |
|
200 |
|
opelxp1 |
|
201 |
|
elsni |
|
202 |
200 201
|
syl |
|
203 |
199 202
|
mto |
|
204 |
|
1onn |
|
205 |
|
nnord |
|
206 |
|
ordirr |
|
207 |
204 205 206
|
mp2b |
|
208 |
|
opelxp2 |
|
209 |
207 208
|
mto |
|
210 |
203 209
|
pm3.2ni |
|
211 |
|
elun |
|
212 |
210 211
|
mtbir |
|
213 |
|
ssnelpss |
|
214 |
197 212 213
|
mp2ani |
|
215 |
189 214
|
syl |
|
216 |
|
df-dju |
|
217 |
|
df-dju |
|
218 |
216 217
|
psseq12i |
|
219 |
215 218
|
sylibr |
|
220 |
|
php3 |
|
221 |
182 219 220
|
syl2anc |
|
222 |
|
canthp1lem1 |
|
223 |
1 222
|
syl |
|
224 |
|
sdomdomtr |
|
225 |
221 223 224
|
syl2anc |
|
226 |
|
sdomnen |
|
227 |
225 226
|
syl |
|
228 |
13 227
|
pm2.65i |
|