Step |
Hyp |
Ref |
Expression |
1 |
|
psmetf |
|
2 |
1
|
adantr |
|
3 |
|
simpr |
|
4 |
|
xpss12 |
|
5 |
3 3 4
|
syl2anc |
|
6 |
2 5
|
fssresd |
|
7 |
|
simpr |
|
8 |
7 7
|
ovresd |
|
9 |
|
simpll |
|
10 |
3
|
sselda |
|
11 |
|
psmet0 |
|
12 |
9 10 11
|
syl2anc |
|
13 |
8 12
|
eqtrd |
|
14 |
9
|
ad2antrr |
|
15 |
3
|
ad2antrr |
|
16 |
15
|
sselda |
|
17 |
10
|
ad2antrr |
|
18 |
3
|
adantr |
|
19 |
18
|
sselda |
|
20 |
19
|
adantr |
|
21 |
|
psmettri2 |
|
22 |
14 16 17 20 21
|
syl13anc |
|
23 |
7
|
adantr |
|
24 |
|
simpr |
|
25 |
23 24
|
ovresd |
|
26 |
25
|
adantr |
|
27 |
|
simpr |
|
28 |
7
|
ad2antrr |
|
29 |
27 28
|
ovresd |
|
30 |
24
|
adantr |
|
31 |
27 30
|
ovresd |
|
32 |
29 31
|
oveq12d |
|
33 |
22 26 32
|
3brtr4d |
|
34 |
33
|
ralrimiva |
|
35 |
34
|
ralrimiva |
|
36 |
13 35
|
jca |
|
37 |
36
|
ralrimiva |
|
38 |
|
elfvex |
|
39 |
38
|
adantr |
|
40 |
39 3
|
ssexd |
|
41 |
|
ispsmet |
|
42 |
40 41
|
syl |
|
43 |
6 37 42
|
mpbir2and |
|