Metamath Proof Explorer


Theorem tskcard

Description: An even more direct relationship than r1tskina to get an inaccessible cardinal out of a Tarski class: the size of any nonempty Tarski class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013)

Ref Expression
Assertion tskcard
|- ( ( T e. Tarski /\ T =/= (/) ) -> ( card ` T ) e. Inacc )

Proof

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 )