Step |
Hyp |
Ref |
Expression |
1 |
|
sdomdom |
|- ( A ~< B -> A ~<_ B ) |
2 |
|
relsdom |
|- Rel ~< |
3 |
2
|
brrelex2i |
|- ( A ~< B -> B e. _V ) |
4 |
|
brdomg |
|- ( B e. _V -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) |
5 |
3 4
|
syl |
|- ( A ~< B -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) |
6 |
1 5
|
mpbid |
|- ( A ~< B -> E. f f : A -1-1-> B ) |
7 |
6
|
adantr |
|- ( ( A ~< B /\ C e. B ) -> E. f f : A -1-1-> B ) |
8 |
|
f1f |
|- ( f : A -1-1-> B -> f : A --> B ) |
9 |
8
|
frnd |
|- ( f : A -1-1-> B -> ran f C_ B ) |
10 |
9
|
adantl |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ran f C_ B ) |
11 |
|
sdomnen |
|- ( A ~< B -> -. A ~~ B ) |
12 |
11
|
ad2antrr |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> -. A ~~ B ) |
13 |
|
vex |
|- f e. _V |
14 |
|
dff1o5 |
|- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) ) |
15 |
14
|
biimpri |
|- ( ( f : A -1-1-> B /\ ran f = B ) -> f : A -1-1-onto-> B ) |
16 |
|
f1oen3g |
|- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B ) |
17 |
13 15 16
|
sylancr |
|- ( ( f : A -1-1-> B /\ ran f = B ) -> A ~~ B ) |
18 |
17
|
ex |
|- ( f : A -1-1-> B -> ( ran f = B -> A ~~ B ) ) |
19 |
18
|
necon3bd |
|- ( f : A -1-1-> B -> ( -. A ~~ B -> ran f =/= B ) ) |
20 |
19
|
adantl |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( -. A ~~ B -> ran f =/= B ) ) |
21 |
12 20
|
mpd |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ran f =/= B ) |
22 |
|
pssdifn0 |
|- ( ( ran f C_ B /\ ran f =/= B ) -> ( B \ ran f ) =/= (/) ) |
23 |
10 21 22
|
syl2anc |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( B \ ran f ) =/= (/) ) |
24 |
|
n0 |
|- ( ( B \ ran f ) =/= (/) <-> E. x x e. ( B \ ran f ) ) |
25 |
23 24
|
sylib |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> E. x x e. ( B \ ran f ) ) |
26 |
2
|
brrelex1i |
|- ( A ~< B -> A e. _V ) |
27 |
26
|
ad2antrr |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A e. _V ) |
28 |
3
|
ad2antrr |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> B e. _V ) |
29 |
|
difexg |
|- ( B e. _V -> ( B \ { x } ) e. _V ) |
30 |
28 29
|
syl |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> ( B \ { x } ) e. _V ) |
31 |
|
eldifn |
|- ( x e. ( B \ ran f ) -> -. x e. ran f ) |
32 |
|
disjsn |
|- ( ( ran f i^i { x } ) = (/) <-> -. x e. ran f ) |
33 |
31 32
|
sylibr |
|- ( x e. ( B \ ran f ) -> ( ran f i^i { x } ) = (/) ) |
34 |
33
|
adantl |
|- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ( ran f i^i { x } ) = (/) ) |
35 |
9
|
adantr |
|- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ran f C_ B ) |
36 |
|
reldisj |
|- ( ran f C_ B -> ( ( ran f i^i { x } ) = (/) <-> ran f C_ ( B \ { x } ) ) ) |
37 |
35 36
|
syl |
|- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ( ( ran f i^i { x } ) = (/) <-> ran f C_ ( B \ { x } ) ) ) |
38 |
34 37
|
mpbid |
|- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> ran f C_ ( B \ { x } ) ) |
39 |
|
f1ssr |
|- ( ( f : A -1-1-> B /\ ran f C_ ( B \ { x } ) ) -> f : A -1-1-> ( B \ { x } ) ) |
40 |
38 39
|
syldan |
|- ( ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) -> f : A -1-1-> ( B \ { x } ) ) |
41 |
40
|
adantl |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> f : A -1-1-> ( B \ { x } ) ) |
42 |
|
f1dom2g |
|- ( ( A e. _V /\ ( B \ { x } ) e. _V /\ f : A -1-1-> ( B \ { x } ) ) -> A ~<_ ( B \ { x } ) ) |
43 |
27 30 41 42
|
syl3anc |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A ~<_ ( B \ { x } ) ) |
44 |
|
eldifi |
|- ( x e. ( B \ ran f ) -> x e. B ) |
45 |
44
|
ad2antll |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> x e. B ) |
46 |
|
simplr |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> C e. B ) |
47 |
|
difsnen |
|- ( ( B e. _V /\ x e. B /\ C e. B ) -> ( B \ { x } ) ~~ ( B \ { C } ) ) |
48 |
28 45 46 47
|
syl3anc |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> ( B \ { x } ) ~~ ( B \ { C } ) ) |
49 |
|
domentr |
|- ( ( A ~<_ ( B \ { x } ) /\ ( B \ { x } ) ~~ ( B \ { C } ) ) -> A ~<_ ( B \ { C } ) ) |
50 |
43 48 49
|
syl2anc |
|- ( ( ( A ~< B /\ C e. B ) /\ ( f : A -1-1-> B /\ x e. ( B \ ran f ) ) ) -> A ~<_ ( B \ { C } ) ) |
51 |
50
|
expr |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( x e. ( B \ ran f ) -> A ~<_ ( B \ { C } ) ) ) |
52 |
51
|
exlimdv |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> ( E. x x e. ( B \ ran f ) -> A ~<_ ( B \ { C } ) ) ) |
53 |
25 52
|
mpd |
|- ( ( ( A ~< B /\ C e. B ) /\ f : A -1-1-> B ) -> A ~<_ ( B \ { C } ) ) |
54 |
7 53
|
exlimddv |
|- ( ( A ~< B /\ C e. B ) -> A ~<_ ( B \ { C } ) ) |
55 |
1
|
adantr |
|- ( ( A ~< B /\ -. C e. B ) -> A ~<_ B ) |
56 |
|
difsn |
|- ( -. C e. B -> ( B \ { C } ) = B ) |
57 |
56
|
breq2d |
|- ( -. C e. B -> ( A ~<_ ( B \ { C } ) <-> A ~<_ B ) ) |
58 |
57
|
adantl |
|- ( ( A ~< B /\ -. C e. B ) -> ( A ~<_ ( B \ { C } ) <-> A ~<_ B ) ) |
59 |
55 58
|
mpbird |
|- ( ( A ~< B /\ -. C e. B ) -> A ~<_ ( B \ { C } ) ) |
60 |
54 59
|
pm2.61dan |
|- ( A ~< B -> A ~<_ ( B \ { C } ) ) |