Step |
Hyp |
Ref |
Expression |
1 |
|
funfn |
|
2 |
|
hashfn |
|
3 |
1 2
|
sylbi |
|
4 |
|
dmfi |
|
5 |
|
hashcl |
|
6 |
4 5
|
syl |
|
7 |
6
|
nn0red |
|
8 |
7
|
adantr |
|
9 |
|
df-rel |
|
10 |
|
dfss3 |
|
11 |
9 10
|
bitri |
|
12 |
11
|
notbii |
|
13 |
|
rexnal |
|
14 |
12 13
|
bitr4i |
|
15 |
|
dmun |
|
16 |
15
|
fveq2i |
|
17 |
|
dmsnn0 |
|
18 |
17
|
biimpri |
|
19 |
18
|
necon1bi |
|
20 |
19
|
3ad2ant3 |
|
21 |
20
|
uneq2d |
|
22 |
|
un0 |
|
23 |
21 22
|
eqtrdi |
|
24 |
23
|
fveq2d |
|
25 |
16 24
|
syl5eq |
|
26 |
|
diffi |
|
27 |
|
dmfi |
|
28 |
26 27
|
syl |
|
29 |
|
hashcl |
|
30 |
28 29
|
syl |
|
31 |
30
|
nn0red |
|
32 |
|
hashcl |
|
33 |
26 32
|
syl |
|
34 |
33
|
nn0red |
|
35 |
|
peano2re |
|
36 |
34 35
|
syl |
|
37 |
|
fidomdm |
|
38 |
26 37
|
syl |
|
39 |
|
hashdom |
|
40 |
28 26 39
|
syl2anc |
|
41 |
38 40
|
mpbird |
|
42 |
34
|
ltp1d |
|
43 |
31 34 36 41 42
|
lelttrd |
|
44 |
43
|
3ad2ant1 |
|
45 |
25 44
|
eqbrtrd |
|
46 |
|
snfi |
|
47 |
|
incom |
|
48 |
|
disjdif |
|
49 |
47 48
|
eqtri |
|
50 |
|
hashun |
|
51 |
46 49 50
|
mp3an23 |
|
52 |
26 51
|
syl |
|
53 |
|
hashsng |
|
54 |
53
|
elv |
|
55 |
54
|
oveq2i |
|
56 |
52 55
|
eqtr2di |
|
57 |
56
|
3ad2ant1 |
|
58 |
45 57
|
breqtrd |
|
59 |
|
difsnid |
|
60 |
59
|
dmeqd |
|
61 |
60
|
fveq2d |
|
62 |
61
|
3ad2ant2 |
|
63 |
59
|
fveq2d |
|
64 |
63
|
3ad2ant2 |
|
65 |
58 62 64
|
3brtr3d |
|
66 |
65
|
rexlimdv3a |
|
67 |
14 66
|
syl5bi |
|
68 |
67
|
imp |
|
69 |
8 68
|
gtned |
|
70 |
69
|
ex |
|
71 |
70
|
necon4bd |
|
72 |
71
|
imp |
|
73 |
|
2nalexn |
|
74 |
|
df-ne |
|
75 |
74
|
anbi2i |
|
76 |
|
annim |
|
77 |
75 76
|
bitri |
|
78 |
77
|
exbii |
|
79 |
|
exnal |
|
80 |
78 79
|
bitr2i |
|
81 |
80
|
2exbii |
|
82 |
73 81
|
bitri |
|
83 |
7
|
adantr |
|
84 |
|
2re |
|
85 |
|
diffi |
|
86 |
|
dmfi |
|
87 |
85 86
|
syl |
|
88 |
|
hashcl |
|
89 |
87 88
|
syl |
|
90 |
89
|
nn0red |
|
91 |
90
|
adantr |
|
92 |
|
readdcl |
|
93 |
84 91 92
|
sylancr |
|
94 |
|
hashcl |
|
95 |
94
|
nn0red |
|
96 |
95
|
adantr |
|
97 |
|
1re |
|
98 |
|
readdcl |
|
99 |
97 90 98
|
sylancr |
|
100 |
99
|
adantr |
|
101 |
84 90 92
|
sylancr |
|
102 |
101
|
adantr |
|
103 |
|
dmun |
|
104 |
|
opex |
|
105 |
|
opex |
|
106 |
104 105
|
prss |
|
107 |
|
undif |
|
108 |
106 107
|
sylbb |
|
109 |
108
|
dmeqd |
|
110 |
103 109
|
syl5reqr |
|
111 |
|
vex |
|
112 |
|
vex |
|
113 |
111 112
|
dmprop |
|
114 |
|
dfsn2 |
|
115 |
113 114
|
eqtr4i |
|
116 |
115
|
uneq1i |
|
117 |
110 116
|
eqtrdi |
|
118 |
117
|
fveq2d |
|
119 |
118
|
ad2antrl |
|
120 |
|
hashun2 |
|
121 |
46 87 120
|
sylancr |
|
122 |
54
|
oveq1i |
|
123 |
121 122
|
breqtrdi |
|
124 |
123
|
adantr |
|
125 |
119 124
|
eqbrtrd |
|
126 |
|
1lt2 |
|
127 |
|
ltadd1 |
|
128 |
97 84 90 127
|
mp3an12i |
|
129 |
126 128
|
mpbii |
|
130 |
129
|
adantr |
|
131 |
83 100 102 125 130
|
lelttrd |
|
132 |
|
fidomdm |
|
133 |
85 132
|
syl |
|
134 |
|
hashdom |
|
135 |
87 85 134
|
syl2anc |
|
136 |
133 135
|
mpbird |
|
137 |
|
hashcl |
|
138 |
85 137
|
syl |
|
139 |
138
|
nn0red |
|
140 |
|
leadd2 |
|
141 |
84 140
|
mp3an3 |
|
142 |
90 139 141
|
syl2anc |
|
143 |
136 142
|
mpbid |
|
144 |
143
|
adantr |
|
145 |
|
prfi |
|
146 |
|
disjdif |
|
147 |
|
hashun |
|
148 |
145 146 147
|
mp3an13 |
|
149 |
85 148
|
syl |
|
150 |
149
|
adantr |
|
151 |
108
|
fveq2d |
|
152 |
151
|
ad2antrl |
|
153 |
|
vex |
|
154 |
153 111
|
opth |
|
155 |
154
|
simprbi |
|
156 |
155
|
necon3i |
|
157 |
|
hashprg |
|
158 |
104 105 157
|
mp2an |
|
159 |
156 158
|
sylib |
|
160 |
159
|
oveq1d |
|
161 |
160
|
ad2antll |
|
162 |
150 152 161
|
3eqtr3rd |
|
163 |
144 162
|
breqtrd |
|
164 |
83 93 96 131 163
|
ltletrd |
|
165 |
83 164
|
gtned |
|
166 |
165
|
ex |
|
167 |
166
|
exlimdv |
|
168 |
167
|
exlimdvv |
|
169 |
82 168
|
syl5bi |
|
170 |
169
|
necon4bd |
|
171 |
170
|
imp |
|
172 |
|
dffun4 |
|
173 |
72 171 172
|
sylanbrc |
|
174 |
173
|
ex |
|
175 |
3 174
|
impbid2 |
|