Step |
Hyp |
Ref |
Expression |
1 |
|
salgensscntex.a |
|- A = ( 0 [,] 2 ) |
2 |
|
salgensscntex.s |
|- S = { x e. ~P A | ( x ~<_ _om \/ ( A \ x ) ~<_ _om ) } |
3 |
|
salgensscntex.x |
|- X = ran ( y e. ( 0 [,] 1 ) |-> { y } ) |
4 |
|
salgensscntex.g |
|- G = ( SalGen ` X ) |
5 |
|
0re |
|- 0 e. RR |
6 |
|
2re |
|- 2 e. RR |
7 |
5 6
|
pm3.2i |
|- ( 0 e. RR /\ 2 e. RR ) |
8 |
5
|
leidi |
|- 0 <_ 0 |
9 |
|
1le2 |
|- 1 <_ 2 |
10 |
8 9
|
pm3.2i |
|- ( 0 <_ 0 /\ 1 <_ 2 ) |
11 |
|
iccss |
|- ( ( ( 0 e. RR /\ 2 e. RR ) /\ ( 0 <_ 0 /\ 1 <_ 2 ) ) -> ( 0 [,] 1 ) C_ ( 0 [,] 2 ) ) |
12 |
7 10 11
|
mp2an |
|- ( 0 [,] 1 ) C_ ( 0 [,] 2 ) |
13 |
|
id |
|- ( y e. ( 0 [,] 1 ) -> y e. ( 0 [,] 1 ) ) |
14 |
12 13
|
sseldi |
|- ( y e. ( 0 [,] 1 ) -> y e. ( 0 [,] 2 ) ) |
15 |
14 1
|
eleqtrrdi |
|- ( y e. ( 0 [,] 1 ) -> y e. A ) |
16 |
|
snelpwi |
|- ( y e. A -> { y } e. ~P A ) |
17 |
15 16
|
syl |
|- ( y e. ( 0 [,] 1 ) -> { y } e. ~P A ) |
18 |
|
snfi |
|- { y } e. Fin |
19 |
|
fict |
|- ( { y } e. Fin -> { y } ~<_ _om ) |
20 |
18 19
|
ax-mp |
|- { y } ~<_ _om |
21 |
|
orc |
|- ( { y } ~<_ _om -> ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) ) |
22 |
20 21
|
ax-mp |
|- ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) |
23 |
22
|
a1i |
|- ( y e. ( 0 [,] 1 ) -> ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) ) |
24 |
17 23
|
jca |
|- ( y e. ( 0 [,] 1 ) -> ( { y } e. ~P A /\ ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) ) ) |
25 |
|
breq1 |
|- ( x = { y } -> ( x ~<_ _om <-> { y } ~<_ _om ) ) |
26 |
|
difeq2 |
|- ( x = { y } -> ( A \ x ) = ( A \ { y } ) ) |
27 |
26
|
breq1d |
|- ( x = { y } -> ( ( A \ x ) ~<_ _om <-> ( A \ { y } ) ~<_ _om ) ) |
28 |
25 27
|
orbi12d |
|- ( x = { y } -> ( ( x ~<_ _om \/ ( A \ x ) ~<_ _om ) <-> ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) ) ) |
29 |
28 2
|
elrab2 |
|- ( { y } e. S <-> ( { y } e. ~P A /\ ( { y } ~<_ _om \/ ( A \ { y } ) ~<_ _om ) ) ) |
30 |
24 29
|
sylibr |
|- ( y e. ( 0 [,] 1 ) -> { y } e. S ) |
31 |
30
|
rgen |
|- A. y e. ( 0 [,] 1 ) { y } e. S |
32 |
|
eqid |
|- ( y e. ( 0 [,] 1 ) |-> { y } ) = ( y e. ( 0 [,] 1 ) |-> { y } ) |
33 |
32
|
rnmptss |
|- ( A. y e. ( 0 [,] 1 ) { y } e. S -> ran ( y e. ( 0 [,] 1 ) |-> { y } ) C_ S ) |
34 |
31 33
|
ax-mp |
|- ran ( y e. ( 0 [,] 1 ) |-> { y } ) C_ S |
35 |
3 34
|
eqsstri |
|- X C_ S |
36 |
|
ovex |
|- ( 0 [,] 2 ) e. _V |
37 |
1 36
|
eqeltri |
|- A e. _V |
38 |
37
|
a1i |
|- ( T. -> A e. _V ) |
39 |
38 2
|
salexct |
|- ( T. -> S e. SAlg ) |
40 |
39
|
mptru |
|- S e. SAlg |
41 |
|
ovex |
|- ( 0 [,] 1 ) e. _V |
42 |
41
|
mptex |
|- ( y e. ( 0 [,] 1 ) |-> { y } ) e. _V |
43 |
42
|
rnex |
|- ran ( y e. ( 0 [,] 1 ) |-> { y } ) e. _V |
44 |
3 43
|
eqeltri |
|- X e. _V |
45 |
44
|
a1i |
|- ( T. -> X e. _V ) |
46 |
3
|
unieqi |
|- U. X = U. ran ( y e. ( 0 [,] 1 ) |-> { y } ) |
47 |
|
snex |
|- { y } e. _V |
48 |
47
|
rgenw |
|- A. y e. ( 0 [,] 1 ) { y } e. _V |
49 |
|
dfiun3g |
|- ( A. y e. ( 0 [,] 1 ) { y } e. _V -> U_ y e. ( 0 [,] 1 ) { y } = U. ran ( y e. ( 0 [,] 1 ) |-> { y } ) ) |
50 |
48 49
|
ax-mp |
|- U_ y e. ( 0 [,] 1 ) { y } = U. ran ( y e. ( 0 [,] 1 ) |-> { y } ) |
51 |
50
|
eqcomi |
|- U. ran ( y e. ( 0 [,] 1 ) |-> { y } ) = U_ y e. ( 0 [,] 1 ) { y } |
52 |
|
iunid |
|- U_ y e. ( 0 [,] 1 ) { y } = ( 0 [,] 1 ) |
53 |
46 51 52
|
3eqtrri |
|- ( 0 [,] 1 ) = U. X |
54 |
45 4 53
|
unisalgen |
|- ( T. -> ( 0 [,] 1 ) e. G ) |
55 |
54
|
mptru |
|- ( 0 [,] 1 ) e. G |
56 |
|
eqid |
|- ( 0 [,] 1 ) = ( 0 [,] 1 ) |
57 |
1 2 56
|
salexct2 |
|- -. ( 0 [,] 1 ) e. S |
58 |
|
nelss |
|- ( ( ( 0 [,] 1 ) e. G /\ -. ( 0 [,] 1 ) e. S ) -> -. G C_ S ) |
59 |
55 57 58
|
mp2an |
|- -. G C_ S |
60 |
35 40 59
|
3pm3.2i |
|- ( X C_ S /\ S e. SAlg /\ -. G C_ S ) |