Step |
Hyp |
Ref |
Expression |
1 |
|
pimxrneun.1 |
|- F/ x ph |
2 |
|
pimxrneun.2 |
|- ( ( ph /\ x e. A ) -> B e. RR* ) |
3 |
|
pimxrneun.3 |
|- ( ( ph /\ x e. A ) -> C e. RR* ) |
4 |
|
nfrab1 |
|- F/_ x { x e. A | B < C } |
5 |
|
nfrab1 |
|- F/_ x { x e. A | C < B } |
6 |
4 5
|
nfun |
|- F/_ x ( { x e. A | B < C } u. { x e. A | C < B } ) |
7 |
|
simpl |
|- ( ( x e. A /\ B < C ) -> x e. A ) |
8 |
|
simpr |
|- ( ( x e. A /\ B < C ) -> B < C ) |
9 |
7 8
|
jca |
|- ( ( x e. A /\ B < C ) -> ( x e. A /\ B < C ) ) |
10 |
|
rabid |
|- ( x e. { x e. A | B < C } <-> ( x e. A /\ B < C ) ) |
11 |
9 10
|
sylibr |
|- ( ( x e. A /\ B < C ) -> x e. { x e. A | B < C } ) |
12 |
11
|
adantll |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> x e. { x e. A | B < C } ) |
13 |
|
elun1 |
|- ( x e. { x e. A | B < C } -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
14 |
12 13
|
syl |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
15 |
14
|
3adantl3 |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ B < C ) -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
16 |
|
3simpa |
|- ( ( ph /\ x e. A /\ B =/= C ) -> ( ph /\ x e. A ) ) |
17 |
16
|
adantr |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> ( ph /\ x e. A ) ) |
18 |
3
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ -. B < C ) -> C e. RR* ) |
19 |
18
|
3adantl3 |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> C e. RR* ) |
20 |
2
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ -. B < C ) -> B e. RR* ) |
21 |
20
|
3adantl3 |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> B e. RR* ) |
22 |
|
simpr |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> -. B < C ) |
23 |
19 21 22
|
xrnltled |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> C <_ B ) |
24 |
|
necom |
|- ( B =/= C <-> C =/= B ) |
25 |
24
|
biimpi |
|- ( B =/= C -> C =/= B ) |
26 |
25
|
adantr |
|- ( ( B =/= C /\ -. B < C ) -> C =/= B ) |
27 |
26
|
3ad2antl3 |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> C =/= B ) |
28 |
19 21 23 27
|
xrleneltd |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> C < B ) |
29 |
|
id |
|- ( ( x e. A /\ C < B ) -> ( x e. A /\ C < B ) ) |
30 |
29
|
adantll |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> ( x e. A /\ C < B ) ) |
31 |
|
rabid |
|- ( x e. { x e. A | C < B } <-> ( x e. A /\ C < B ) ) |
32 |
30 31
|
sylibr |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> x e. { x e. A | C < B } ) |
33 |
|
elun2 |
|- ( x e. { x e. A | C < B } -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
34 |
32 33
|
syl |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
35 |
17 28 34
|
syl2anc |
|- ( ( ( ph /\ x e. A /\ B =/= C ) /\ -. B < C ) -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
36 |
15 35
|
pm2.61dan |
|- ( ( ph /\ x e. A /\ B =/= C ) -> x e. ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
37 |
1 6 36
|
rabssd |
|- ( ph -> { x e. A | B =/= C } C_ ( { x e. A | B < C } u. { x e. A | C < B } ) ) |
38 |
2
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> B e. RR* ) |
39 |
3
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> C e. RR* ) |
40 |
|
simpr |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> B < C ) |
41 |
38 39 40
|
xrltned |
|- ( ( ( ph /\ x e. A ) /\ B < C ) -> B =/= C ) |
42 |
41
|
ex |
|- ( ( ph /\ x e. A ) -> ( B < C -> B =/= C ) ) |
43 |
1 42
|
ss2rabdf |
|- ( ph -> { x e. A | B < C } C_ { x e. A | B =/= C } ) |
44 |
3
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> C e. RR* ) |
45 |
2
|
adantr |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> B e. RR* ) |
46 |
|
simpr |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> C < B ) |
47 |
44 45 46
|
xrgtned |
|- ( ( ( ph /\ x e. A ) /\ C < B ) -> B =/= C ) |
48 |
47
|
ex |
|- ( ( ph /\ x e. A ) -> ( C < B -> B =/= C ) ) |
49 |
1 48
|
ss2rabdf |
|- ( ph -> { x e. A | C < B } C_ { x e. A | B =/= C } ) |
50 |
43 49
|
unssd |
|- ( ph -> ( { x e. A | B < C } u. { x e. A | C < B } ) C_ { x e. A | B =/= C } ) |
51 |
37 50
|
eqssd |
|- ( ph -> { x e. A | B =/= C } = ( { x e. A | B < C } u. { x e. A | C < B } ) ) |