Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
|
2 |
1
|
adantl |
|
3 |
|
simpll |
|
4 |
|
simplr |
|
5 |
2 3 4
|
3jca |
|
6 |
|
elex |
|
7 |
6
|
3anim1i |
|
8 |
7
|
3expib |
|
9 |
|
elex |
|
10 |
9
|
3anim1i |
|
11 |
10
|
3expib |
|
12 |
|
vex |
|
13 |
12
|
inex1 |
|
14 |
|
eleq1 |
|
15 |
13 14
|
mpbiri |
|
16 |
15
|
a1i |
|
17 |
16
|
rexlimivv |
|
18 |
17
|
3anim1i |
|
19 |
18
|
3expib |
|
20 |
8 11 19
|
3jaoi |
|
21 |
20
|
impcom |
|
22 |
|
simp1 |
|
23 |
|
unexg |
|
24 |
23
|
3adant1 |
|
25 |
|
elfi |
|
26 |
22 24 25
|
syl2anc |
|
27 |
|
simpl1 |
|
28 |
|
eleq1 |
|
29 |
|
intex |
|
30 |
28 29
|
bitr4di |
|
31 |
27 30
|
syl5ibcom |
|
32 |
|
simp22 |
|
33 |
|
inss2 |
|
34 |
33
|
a1i |
|
35 |
|
simp1l |
|
36 |
|
simp3l |
|
37 |
36
|
elin2d |
|
38 |
|
inss1 |
|
39 |
|
ssfi |
|
40 |
37 38 39
|
sylancl |
|
41 |
|
elfir |
|
42 |
32 34 35 40 41
|
syl13anc |
|
43 |
|
simp23 |
|
44 |
|
inss2 |
|
45 |
44
|
a1i |
|
46 |
|
simp1r |
|
47 |
|
inss1 |
|
48 |
|
ssfi |
|
49 |
37 47 48
|
sylancl |
|
50 |
|
elfir |
|
51 |
43 45 46 49 50
|
syl13anc |
|
52 |
|
elinel1 |
|
53 |
52
|
elpwid |
|
54 |
|
indi |
|
55 |
|
df-ss |
|
56 |
55
|
biimpi |
|
57 |
54 56
|
syl5reqr |
|
58 |
57
|
inteqd |
|
59 |
|
intun |
|
60 |
58 59
|
eqtrdi |
|
61 |
36 53 60
|
3syl |
|
62 |
|
ineq1 |
|
63 |
62
|
eqeq2d |
|
64 |
|
ineq2 |
|
65 |
64
|
eqeq2d |
|
66 |
63 65
|
rspc2ev |
|
67 |
42 51 61 66
|
syl3anc |
|
68 |
67
|
3mix3d |
|
69 |
68
|
3expib |
|
70 |
|
simp23 |
|
71 |
|
simp1 |
|
72 |
|
simp3l |
|
73 |
|
reldisj |
|
74 |
72 53 73
|
3syl |
|
75 |
71 74
|
mpbid |
|
76 |
|
uncom |
|
77 |
76
|
difeq1i |
|
78 |
|
difun2 |
|
79 |
77 78
|
eqtri |
|
80 |
|
difss |
|
81 |
79 80
|
eqsstri |
|
82 |
75 81
|
sstrdi |
|
83 |
|
simp3r |
|
84 |
72
|
elin2d |
|
85 |
|
elfir |
|
86 |
70 82 83 84 85
|
syl13anc |
|
87 |
86
|
3mix2d |
|
88 |
87
|
3expib |
|
89 |
|
simp22 |
|
90 |
|
simp1 |
|
91 |
|
simp3l |
|
92 |
|
reldisj |
|
93 |
91 53 92
|
3syl |
|
94 |
90 93
|
mpbid |
|
95 |
|
difun2 |
|
96 |
|
difss |
|
97 |
95 96
|
eqsstri |
|
98 |
94 97
|
sstrdi |
|
99 |
|
simp3r |
|
100 |
91
|
elin2d |
|
101 |
|
elfir |
|
102 |
89 98 99 100 101
|
syl13anc |
|
103 |
102
|
3mix1d |
|
104 |
103
|
3expib |
|
105 |
69 88 104
|
pm2.61iine |
|
106 |
|
eleq1 |
|
107 |
|
eleq1 |
|
108 |
|
eqeq1 |
|
109 |
108
|
2rexbidv |
|
110 |
106 107 109
|
3orbi123d |
|
111 |
105 110
|
syl5ibrcom |
|
112 |
111
|
expr |
|
113 |
112
|
com23 |
|
114 |
31 113
|
mpdd |
|
115 |
114
|
rexlimdva |
|
116 |
26 115
|
sylbid |
|
117 |
|
ssun1 |
|
118 |
|
fiss |
|
119 |
23 117 118
|
sylancl |
|
120 |
119
|
3adant1 |
|
121 |
120
|
sseld |
|
122 |
|
ssun2 |
|
123 |
|
fiss |
|
124 |
23 122 123
|
sylancl |
|
125 |
124
|
3adant1 |
|
126 |
125
|
sseld |
|
127 |
120
|
sseld |
|
128 |
125
|
sseld |
|
129 |
127 128
|
anim12d |
|
130 |
|
fiin |
|
131 |
|
eleq1a |
|
132 |
130 131
|
syl |
|
133 |
129 132
|
syl6 |
|
134 |
133
|
rexlimdvv |
|
135 |
121 126 134
|
3jaod |
|
136 |
116 135
|
impbid |
|
137 |
5 21 136
|
pm5.21nd |
|