Step |
Hyp |
Ref |
Expression |
1 |
|
cardeq0 |
|- ( T e. Tarski -> ( ( card ` T ) = (/) <-> T = (/) ) ) |
2 |
1
|
necon3bid |
|- ( T e. Tarski -> ( ( card ` T ) =/= (/) <-> T =/= (/) ) ) |
3 |
2
|
biimpar |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( card ` T ) =/= (/) ) |
4 |
|
eqid |
|- ( z e. ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) |-> ( har ` ( w ` z ) ) ) = ( z e. ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) |-> ( har ` ( w ` z ) ) ) |
5 |
4
|
pwcfsdom |
|- ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ~< ( ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ^m ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) ) |
6 |
|
vpwex |
|- ~P x e. _V |
7 |
6
|
canth2 |
|- ~P x ~< ~P ~P x |
8 |
|
simpl |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> T e. Tarski ) |
9 |
|
cardon |
|- ( card ` T ) e. On |
10 |
9
|
oneli |
|- ( x e. ( card ` T ) -> x e. On ) |
11 |
10
|
adantl |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> x e. On ) |
12 |
|
cardsdomelir |
|- ( x e. ( card ` T ) -> x ~< T ) |
13 |
12
|
adantl |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> x ~< T ) |
14 |
|
tskord |
|- ( ( T e. Tarski /\ x e. On /\ x ~< T ) -> x e. T ) |
15 |
8 11 13 14
|
syl3anc |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> x e. T ) |
16 |
|
tskpw |
|- ( ( T e. Tarski /\ x e. T ) -> ~P x e. T ) |
17 |
|
tskpwss |
|- ( ( T e. Tarski /\ ~P x e. T ) -> ~P ~P x C_ T ) |
18 |
16 17
|
syldan |
|- ( ( T e. Tarski /\ x e. T ) -> ~P ~P x C_ T ) |
19 |
15 18
|
syldan |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> ~P ~P x C_ T ) |
20 |
|
ssdomg |
|- ( T e. Tarski -> ( ~P ~P x C_ T -> ~P ~P x ~<_ T ) ) |
21 |
8 19 20
|
sylc |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> ~P ~P x ~<_ T ) |
22 |
|
cardidg |
|- ( T e. Tarski -> ( card ` T ) ~~ T ) |
23 |
22
|
ensymd |
|- ( T e. Tarski -> T ~~ ( card ` T ) ) |
24 |
23
|
adantr |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> T ~~ ( card ` T ) ) |
25 |
|
domentr |
|- ( ( ~P ~P x ~<_ T /\ T ~~ ( card ` T ) ) -> ~P ~P x ~<_ ( card ` T ) ) |
26 |
21 24 25
|
syl2anc |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> ~P ~P x ~<_ ( card ` T ) ) |
27 |
|
sdomdomtr |
|- ( ( ~P x ~< ~P ~P x /\ ~P ~P x ~<_ ( card ` T ) ) -> ~P x ~< ( card ` T ) ) |
28 |
7 26 27
|
sylancr |
|- ( ( T e. Tarski /\ x e. ( card ` T ) ) -> ~P x ~< ( card ` T ) ) |
29 |
28
|
ralrimiva |
|- ( T e. Tarski -> A. x e. ( card ` T ) ~P x ~< ( card ` T ) ) |
30 |
29
|
adantr |
|- ( ( T e. Tarski /\ T =/= (/) ) -> A. x e. ( card ` T ) ~P x ~< ( card ` T ) ) |
31 |
|
inawinalem |
|- ( ( card ` T ) e. On -> ( A. x e. ( card ` T ) ~P x ~< ( card ` T ) -> A. x e. ( card ` T ) E. y e. ( card ` T ) x ~< y ) ) |
32 |
9 31
|
ax-mp |
|- ( A. x e. ( card ` T ) ~P x ~< ( card ` T ) -> A. x e. ( card ` T ) E. y e. ( card ` T ) x ~< y ) |
33 |
|
winainflem |
|- ( ( ( card ` T ) =/= (/) /\ ( card ` T ) e. On /\ A. x e. ( card ` T ) E. y e. ( card ` T ) x ~< y ) -> _om C_ ( card ` T ) ) |
34 |
9 33
|
mp3an2 |
|- ( ( ( card ` T ) =/= (/) /\ A. x e. ( card ` T ) E. y e. ( card ` T ) x ~< y ) -> _om C_ ( card ` T ) ) |
35 |
32 34
|
sylan2 |
|- ( ( ( card ` T ) =/= (/) /\ A. x e. ( card ` T ) ~P x ~< ( card ` T ) ) -> _om C_ ( card ` T ) ) |
36 |
3 30 35
|
syl2anc |
|- ( ( T e. Tarski /\ T =/= (/) ) -> _om C_ ( card ` T ) ) |
37 |
|
cardidm |
|- ( card ` ( card ` T ) ) = ( card ` T ) |
38 |
|
cardaleph |
|- ( ( _om C_ ( card ` T ) /\ ( card ` ( card ` T ) ) = ( card ` T ) ) -> ( card ` T ) = ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) |
39 |
36 37 38
|
sylancl |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( card ` T ) = ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) |
40 |
39
|
fveq2d |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( cf ` ( card ` T ) ) = ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) ) |
41 |
39 40
|
oveq12d |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) = ( ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ^m ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) ) ) |
42 |
39 41
|
breq12d |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) <-> ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ~< ( ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ^m ( cf ` ( aleph ` |^| { x e. On | ( card ` T ) C_ ( aleph ` x ) } ) ) ) ) ) |
43 |
5 42
|
mpbiri |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) |
44 |
|
simp1 |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> T e. Tarski ) |
45 |
|
simp3 |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) |
46 |
|
fvex |
|- ( card ` T ) e. _V |
47 |
|
fvex |
|- ( cf ` ( card ` T ) ) e. _V |
48 |
46 47
|
elmap |
|- ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) <-> x : ( cf ` ( card ` T ) ) --> ( card ` T ) ) |
49 |
|
fssxp |
|- ( x : ( cf ` ( card ` T ) ) --> ( card ` T ) -> x C_ ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) ) |
50 |
48 49
|
sylbi |
|- ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) -> x C_ ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) ) |
51 |
15
|
ex |
|- ( T e. Tarski -> ( x e. ( card ` T ) -> x e. T ) ) |
52 |
51
|
ssrdv |
|- ( T e. Tarski -> ( card ` T ) C_ T ) |
53 |
|
cfle |
|- ( cf ` ( card ` T ) ) C_ ( card ` T ) |
54 |
|
sstr |
|- ( ( ( cf ` ( card ` T ) ) C_ ( card ` T ) /\ ( card ` T ) C_ T ) -> ( cf ` ( card ` T ) ) C_ T ) |
55 |
53 54
|
mpan |
|- ( ( card ` T ) C_ T -> ( cf ` ( card ` T ) ) C_ T ) |
56 |
|
tskxpss |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) C_ T /\ ( card ` T ) C_ T ) -> ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T ) |
57 |
56
|
3exp |
|- ( T e. Tarski -> ( ( cf ` ( card ` T ) ) C_ T -> ( ( card ` T ) C_ T -> ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T ) ) ) |
58 |
57
|
com23 |
|- ( T e. Tarski -> ( ( card ` T ) C_ T -> ( ( cf ` ( card ` T ) ) C_ T -> ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T ) ) ) |
59 |
55 58
|
mpdi |
|- ( T e. Tarski -> ( ( card ` T ) C_ T -> ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T ) ) |
60 |
52 59
|
mpd |
|- ( T e. Tarski -> ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T ) |
61 |
|
sstr2 |
|- ( x C_ ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) -> ( ( ( cf ` ( card ` T ) ) X. ( card ` T ) ) C_ T -> x C_ T ) ) |
62 |
50 60 61
|
syl2im |
|- ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) -> ( T e. Tarski -> x C_ T ) ) |
63 |
45 44 62
|
sylc |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> x C_ T ) |
64 |
|
simp2 |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> ( cf ` ( card ` T ) ) e. ( card ` T ) ) |
65 |
|
ffn |
|- ( x : ( cf ` ( card ` T ) ) --> ( card ` T ) -> x Fn ( cf ` ( card ` T ) ) ) |
66 |
|
fndmeng |
|- ( ( x Fn ( cf ` ( card ` T ) ) /\ ( cf ` ( card ` T ) ) e. _V ) -> ( cf ` ( card ` T ) ) ~~ x ) |
67 |
65 47 66
|
sylancl |
|- ( x : ( cf ` ( card ` T ) ) --> ( card ` T ) -> ( cf ` ( card ` T ) ) ~~ x ) |
68 |
48 67
|
sylbi |
|- ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) -> ( cf ` ( card ` T ) ) ~~ x ) |
69 |
68
|
ensymd |
|- ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) -> x ~~ ( cf ` ( card ` T ) ) ) |
70 |
|
cardsdomelir |
|- ( ( cf ` ( card ` T ) ) e. ( card ` T ) -> ( cf ` ( card ` T ) ) ~< T ) |
71 |
|
ensdomtr |
|- ( ( x ~~ ( cf ` ( card ` T ) ) /\ ( cf ` ( card ` T ) ) ~< T ) -> x ~< T ) |
72 |
69 70 71
|
syl2an |
|- ( ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> x ~< T ) |
73 |
45 64 72
|
syl2anc |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> x ~< T ) |
74 |
|
tskssel |
|- ( ( T e. Tarski /\ x C_ T /\ x ~< T ) -> x e. T ) |
75 |
44 63 73 74
|
syl3anc |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) /\ x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) -> x e. T ) |
76 |
75
|
3expia |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> ( x e. ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) -> x e. T ) ) |
77 |
76
|
ssrdv |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) C_ T ) |
78 |
|
ssdomg |
|- ( T e. Tarski -> ( ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) C_ T -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ T ) ) |
79 |
78
|
imp |
|- ( ( T e. Tarski /\ ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) C_ T ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ T ) |
80 |
77 79
|
syldan |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ T ) |
81 |
23
|
adantr |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> T ~~ ( card ` T ) ) |
82 |
|
domentr |
|- ( ( ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ T /\ T ~~ ( card ` T ) ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ ( card ` T ) ) |
83 |
80 81 82
|
syl2anc |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ ( card ` T ) ) |
84 |
|
domnsym |
|- ( ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ~<_ ( card ` T ) -> -. ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) |
85 |
83 84
|
syl |
|- ( ( T e. Tarski /\ ( cf ` ( card ` T ) ) e. ( card ` T ) ) -> -. ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) |
86 |
85
|
ex |
|- ( T e. Tarski -> ( ( cf ` ( card ` T ) ) e. ( card ` T ) -> -. ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) ) |
87 |
86
|
adantr |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( ( cf ` ( card ` T ) ) e. ( card ` T ) -> -. ( card ` T ) ~< ( ( card ` T ) ^m ( cf ` ( card ` T ) ) ) ) ) |
88 |
43 87
|
mt2d |
|- ( ( T e. Tarski /\ T =/= (/) ) -> -. ( cf ` ( card ` T ) ) e. ( card ` T ) ) |
89 |
|
cfon |
|- ( cf ` ( card ` T ) ) e. On |
90 |
89 9
|
onsseli |
|- ( ( cf ` ( card ` T ) ) C_ ( card ` T ) <-> ( ( cf ` ( card ` T ) ) e. ( card ` T ) \/ ( cf ` ( card ` T ) ) = ( card ` T ) ) ) |
91 |
53 90
|
mpbi |
|- ( ( cf ` ( card ` T ) ) e. ( card ` T ) \/ ( cf ` ( card ` T ) ) = ( card ` T ) ) |
92 |
91
|
ori |
|- ( -. ( cf ` ( card ` T ) ) e. ( card ` T ) -> ( cf ` ( card ` T ) ) = ( card ` T ) ) |
93 |
88 92
|
syl |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( cf ` ( card ` T ) ) = ( card ` T ) ) |
94 |
|
elina |
|- ( ( card ` T ) e. Inacc <-> ( ( card ` T ) =/= (/) /\ ( cf ` ( card ` T ) ) = ( card ` T ) /\ A. x e. ( card ` T ) ~P x ~< ( card ` T ) ) ) |
95 |
3 93 30 94
|
syl3anbrc |
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( card ` T ) e. Inacc ) |