Step |
Hyp |
Ref |
Expression |
1 |
|
reldom |
|- Rel ~<_ |
2 |
1
|
brrelex2i |
|- ( _om ~<_ A -> A e. _V ) |
3 |
|
djudoml |
|- ( ( A e. _V /\ A e. _V ) -> A ~<_ ( A |_| A ) ) |
4 |
2 2 3
|
syl2anc |
|- ( _om ~<_ A -> A ~<_ ( A |_| A ) ) |
5 |
|
domtr |
|- ( ( _om ~<_ A /\ A ~<_ ( A |_| A ) ) -> _om ~<_ ( A |_| A ) ) |
6 |
4 5
|
mpdan |
|- ( _om ~<_ A -> _om ~<_ ( A |_| A ) ) |
7 |
1
|
brrelex2i |
|- ( _om ~<_ ( A |_| A ) -> ( A |_| A ) e. _V ) |
8 |
|
anidm |
|- ( ( A e. _V /\ A e. _V ) <-> A e. _V ) |
9 |
|
djuexb |
|- ( ( A e. _V /\ A e. _V ) <-> ( A |_| A ) e. _V ) |
10 |
8 9
|
bitr3i |
|- ( A e. _V <-> ( A |_| A ) e. _V ) |
11 |
7 10
|
sylibr |
|- ( _om ~<_ ( A |_| A ) -> A e. _V ) |
12 |
|
domeng |
|- ( ( A |_| A ) e. _V -> ( _om ~<_ ( A |_| A ) <-> E. x ( _om ~~ x /\ x C_ ( A |_| A ) ) ) ) |
13 |
7 12
|
syl |
|- ( _om ~<_ ( A |_| A ) -> ( _om ~<_ ( A |_| A ) <-> E. x ( _om ~~ x /\ x C_ ( A |_| A ) ) ) ) |
14 |
13
|
ibi |
|- ( _om ~<_ ( A |_| A ) -> E. x ( _om ~~ x /\ x C_ ( A |_| A ) ) ) |
15 |
|
indi |
|- ( x i^i ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) ) = ( ( x i^i ( { (/) } X. A ) ) u. ( x i^i ( { 1o } X. A ) ) ) |
16 |
|
simpr |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> x C_ ( A |_| A ) ) |
17 |
|
df-dju |
|- ( A |_| A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) |
18 |
16 17
|
sseqtrdi |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> x C_ ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) ) |
19 |
|
df-ss |
|- ( x C_ ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) <-> ( x i^i ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) ) = x ) |
20 |
18 19
|
sylib |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> ( x i^i ( ( { (/) } X. A ) u. ( { 1o } X. A ) ) ) = x ) |
21 |
15 20
|
eqtr3id |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> ( ( x i^i ( { (/) } X. A ) ) u. ( x i^i ( { 1o } X. A ) ) ) = x ) |
22 |
|
ensym |
|- ( _om ~~ x -> x ~~ _om ) |
23 |
22
|
adantr |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> x ~~ _om ) |
24 |
21 23
|
eqbrtrd |
|- ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> ( ( x i^i ( { (/) } X. A ) ) u. ( x i^i ( { 1o } X. A ) ) ) ~~ _om ) |
25 |
|
cdainflem |
|- ( ( ( x i^i ( { (/) } X. A ) ) u. ( x i^i ( { 1o } X. A ) ) ) ~~ _om -> ( ( x i^i ( { (/) } X. A ) ) ~~ _om \/ ( x i^i ( { 1o } X. A ) ) ~~ _om ) ) |
26 |
|
snex |
|- { (/) } e. _V |
27 |
|
xpexg |
|- ( ( { (/) } e. _V /\ A e. _V ) -> ( { (/) } X. A ) e. _V ) |
28 |
26 27
|
mpan |
|- ( A e. _V -> ( { (/) } X. A ) e. _V ) |
29 |
|
inss2 |
|- ( x i^i ( { (/) } X. A ) ) C_ ( { (/) } X. A ) |
30 |
|
ssdomg |
|- ( ( { (/) } X. A ) e. _V -> ( ( x i^i ( { (/) } X. A ) ) C_ ( { (/) } X. A ) -> ( x i^i ( { (/) } X. A ) ) ~<_ ( { (/) } X. A ) ) ) |
31 |
28 29 30
|
mpisyl |
|- ( A e. _V -> ( x i^i ( { (/) } X. A ) ) ~<_ ( { (/) } X. A ) ) |
32 |
|
0ex |
|- (/) e. _V |
33 |
|
xpsnen2g |
|- ( ( (/) e. _V /\ A e. _V ) -> ( { (/) } X. A ) ~~ A ) |
34 |
32 33
|
mpan |
|- ( A e. _V -> ( { (/) } X. A ) ~~ A ) |
35 |
|
domentr |
|- ( ( ( x i^i ( { (/) } X. A ) ) ~<_ ( { (/) } X. A ) /\ ( { (/) } X. A ) ~~ A ) -> ( x i^i ( { (/) } X. A ) ) ~<_ A ) |
36 |
31 34 35
|
syl2anc |
|- ( A e. _V -> ( x i^i ( { (/) } X. A ) ) ~<_ A ) |
37 |
|
domen1 |
|- ( ( x i^i ( { (/) } X. A ) ) ~~ _om -> ( ( x i^i ( { (/) } X. A ) ) ~<_ A <-> _om ~<_ A ) ) |
38 |
36 37
|
syl5ibcom |
|- ( A e. _V -> ( ( x i^i ( { (/) } X. A ) ) ~~ _om -> _om ~<_ A ) ) |
39 |
|
snex |
|- { 1o } e. _V |
40 |
|
xpexg |
|- ( ( { 1o } e. _V /\ A e. _V ) -> ( { 1o } X. A ) e. _V ) |
41 |
39 40
|
mpan |
|- ( A e. _V -> ( { 1o } X. A ) e. _V ) |
42 |
|
inss2 |
|- ( x i^i ( { 1o } X. A ) ) C_ ( { 1o } X. A ) |
43 |
|
ssdomg |
|- ( ( { 1o } X. A ) e. _V -> ( ( x i^i ( { 1o } X. A ) ) C_ ( { 1o } X. A ) -> ( x i^i ( { 1o } X. A ) ) ~<_ ( { 1o } X. A ) ) ) |
44 |
41 42 43
|
mpisyl |
|- ( A e. _V -> ( x i^i ( { 1o } X. A ) ) ~<_ ( { 1o } X. A ) ) |
45 |
|
1on |
|- 1o e. On |
46 |
|
xpsnen2g |
|- ( ( 1o e. On /\ A e. _V ) -> ( { 1o } X. A ) ~~ A ) |
47 |
45 46
|
mpan |
|- ( A e. _V -> ( { 1o } X. A ) ~~ A ) |
48 |
|
domentr |
|- ( ( ( x i^i ( { 1o } X. A ) ) ~<_ ( { 1o } X. A ) /\ ( { 1o } X. A ) ~~ A ) -> ( x i^i ( { 1o } X. A ) ) ~<_ A ) |
49 |
44 47 48
|
syl2anc |
|- ( A e. _V -> ( x i^i ( { 1o } X. A ) ) ~<_ A ) |
50 |
|
domen1 |
|- ( ( x i^i ( { 1o } X. A ) ) ~~ _om -> ( ( x i^i ( { 1o } X. A ) ) ~<_ A <-> _om ~<_ A ) ) |
51 |
49 50
|
syl5ibcom |
|- ( A e. _V -> ( ( x i^i ( { 1o } X. A ) ) ~~ _om -> _om ~<_ A ) ) |
52 |
38 51
|
jaod |
|- ( A e. _V -> ( ( ( x i^i ( { (/) } X. A ) ) ~~ _om \/ ( x i^i ( { 1o } X. A ) ) ~~ _om ) -> _om ~<_ A ) ) |
53 |
25 52
|
syl5 |
|- ( A e. _V -> ( ( ( x i^i ( { (/) } X. A ) ) u. ( x i^i ( { 1o } X. A ) ) ) ~~ _om -> _om ~<_ A ) ) |
54 |
24 53
|
syl5 |
|- ( A e. _V -> ( ( _om ~~ x /\ x C_ ( A |_| A ) ) -> _om ~<_ A ) ) |
55 |
54
|
exlimdv |
|- ( A e. _V -> ( E. x ( _om ~~ x /\ x C_ ( A |_| A ) ) -> _om ~<_ A ) ) |
56 |
11 14 55
|
sylc |
|- ( _om ~<_ ( A |_| A ) -> _om ~<_ A ) |
57 |
6 56
|
impbii |
|- ( _om ~<_ A <-> _om ~<_ ( A |_| A ) ) |