| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sdom0 |
|- -. A ~< (/) |
| 2 |
|
breq2 |
|- ( B = (/) -> ( A ~< B <-> A ~< (/) ) ) |
| 3 |
1 2
|
mtbiri |
|- ( B = (/) -> -. A ~< B ) |
| 4 |
3
|
con2i |
|- ( A ~< B -> -. B = (/) ) |
| 5 |
|
neq0 |
|- ( -. B = (/) <-> E. z z e. B ) |
| 6 |
4 5
|
sylib |
|- ( A ~< B -> E. z z e. B ) |
| 7 |
|
domdifsn |
|- ( A ~< B -> A ~<_ ( B \ { z } ) ) |
| 8 |
7
|
adantr |
|- ( ( A ~< B /\ z e. B ) -> A ~<_ ( B \ { z } ) ) |
| 9 |
|
en2sn |
|- ( ( C e. _V /\ z e. _V ) -> { C } ~~ { z } ) |
| 10 |
9
|
elvd |
|- ( C e. _V -> { C } ~~ { z } ) |
| 11 |
|
endom |
|- ( { C } ~~ { z } -> { C } ~<_ { z } ) |
| 12 |
10 11
|
syl |
|- ( C e. _V -> { C } ~<_ { z } ) |
| 13 |
|
snprc |
|- ( -. C e. _V <-> { C } = (/) ) |
| 14 |
13
|
biimpi |
|- ( -. C e. _V -> { C } = (/) ) |
| 15 |
|
vsnex |
|- { z } e. _V |
| 16 |
15
|
0dom |
|- (/) ~<_ { z } |
| 17 |
14 16
|
eqbrtrdi |
|- ( -. C e. _V -> { C } ~<_ { z } ) |
| 18 |
12 17
|
pm2.61i |
|- { C } ~<_ { z } |
| 19 |
|
disjdifr |
|- ( ( B \ { z } ) i^i { z } ) = (/) |
| 20 |
|
undom |
|- ( ( ( A ~<_ ( B \ { z } ) /\ { C } ~<_ { z } ) /\ ( ( B \ { z } ) i^i { z } ) = (/) ) -> ( A u. { C } ) ~<_ ( ( B \ { z } ) u. { z } ) ) |
| 21 |
19 20
|
mpan2 |
|- ( ( A ~<_ ( B \ { z } ) /\ { C } ~<_ { z } ) -> ( A u. { C } ) ~<_ ( ( B \ { z } ) u. { z } ) ) |
| 22 |
8 18 21
|
sylancl |
|- ( ( A ~< B /\ z e. B ) -> ( A u. { C } ) ~<_ ( ( B \ { z } ) u. { z } ) ) |
| 23 |
|
uncom |
|- ( ( B \ { z } ) u. { z } ) = ( { z } u. ( B \ { z } ) ) |
| 24 |
|
simpr |
|- ( ( A ~< B /\ z e. B ) -> z e. B ) |
| 25 |
24
|
snssd |
|- ( ( A ~< B /\ z e. B ) -> { z } C_ B ) |
| 26 |
|
undif |
|- ( { z } C_ B <-> ( { z } u. ( B \ { z } ) ) = B ) |
| 27 |
25 26
|
sylib |
|- ( ( A ~< B /\ z e. B ) -> ( { z } u. ( B \ { z } ) ) = B ) |
| 28 |
23 27
|
eqtrid |
|- ( ( A ~< B /\ z e. B ) -> ( ( B \ { z } ) u. { z } ) = B ) |
| 29 |
22 28
|
breqtrd |
|- ( ( A ~< B /\ z e. B ) -> ( A u. { C } ) ~<_ B ) |
| 30 |
6 29
|
exlimddv |
|- ( A ~< B -> ( A u. { C } ) ~<_ B ) |