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Theorem tskcard 9195
Description: An even more direct relationship than r1tskina 9196 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.)
Assertion
Ref Expression
tskcard  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e. 
Inacc )

Proof of Theorem tskcard
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardeq0 8966 . . . 4  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =  (/)  <->  T  =  (/) ) )
21necon3bid 2680 . . 3  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =/=  (/)  <->  T  =/=  (/) ) )
32biimpar 487 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =/=  (/) )
4 eqid 2420 . . . . . 6  |-  ( 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 )
) )
54pwcfsdom 8997 . . . . 5  |-  ( 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 vex 3081 . . . . . . . . . . . . 13  |-  x  e. 
_V
76pwex 4599 . . . . . . . . . . . 12  |-  ~P x  e.  _V
87canth2 7722 . . . . . . . . . . 11  |-  ~P x  ~<  ~P ~P x
9 simpl 458 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  e.  Tarski )
10 cardon 8368 . . . . . . . . . . . . . . . . 17  |-  ( card `  T )  e.  On
1110oneli 5540 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  e.  On )
1211adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  On )
13 cardsdomelir 8397 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  ~<  T )
1413adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  ~<  T )
15 tskord 9194 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  On  /\  x  ~<  T )  ->  x  e.  T )
169, 12, 14, 15syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  T )
17 tskpw 9167 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
18 tskpwss 9166 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  ~P x  e.  T )  ->  ~P ~P x  C_  T )
1917, 18syldan 472 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P ~P x  C_  T )
2016, 19syldan 472 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  C_  T )
21 ssdomg 7613 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  ( ~P ~P x  C_  T  ->  ~P ~P x  ~<_  T )
)
229, 20, 21sylc 62 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  T )
23 cardidg 8962 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( card `  T
)  ~~  T )
2423ensymd 7618 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  T  ~~  ( card `  T ) )
2524adantr 466 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  ~~  ( card `  T
) )
26 domentr 7626 . . . . . . . . . . . 12  |-  ( ( ~P ~P x  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ~P ~P x  ~<_  ( card `  T )
)
2722, 25, 26syl2anc 665 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  ( card `  T ) )
28 sdomdomtr 7702 . . . . . . . . . . 11  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  ( card `  T )
)  ->  ~P x  ~<  ( card `  T
) )
298, 27, 28sylancr 667 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P x  ~<  ( card `  T
) )
3029ralrimiva 2837 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  A. x  e.  (
card `  T ) ~P x  ~<  ( card `  T ) )
3130adantr 466 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  A. x  e.  ( card `  T
) ~P x  ~<  (
card `  T )
)
32 inawinalem 9103 . . . . . . . . . 10  |-  ( (
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
) )
3310, 32ax-mp 5 . . . . . . . . 9  |-  ( A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
)  ->  A. x  e.  ( card `  T
) E. y  e.  ( card `  T
) x  ~<  y
)
34 winainflem 9107 . . . . . . . . . 10  |-  ( ( ( card `  T
)  =/=  (/)  /\  ( card `  T )  e.  On  /\  A. x  e.  ( card `  T
) E. y  e.  ( card `  T
) x  ~<  y
)  ->  om  C_  ( card `  T ) )
3510, 34mp3an2 1348 . . . . . . . . 9  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) E. y  e.  ( card `  T
) x  ~<  y
)  ->  om  C_  ( card `  T ) )
3633, 35sylan2 476 . . . . . . . 8  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
) )  ->  om  C_  ( card `  T ) )
373, 31, 36syl2anc 665 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om  C_  ( card `  T ) )
38 cardidm 8383 . . . . . . 7  |-  ( card `  ( card `  T
) )  =  (
card `  T )
39 cardaleph 8509 . . . . . . 7  |-  ( ( om  C_  ( card `  T )  /\  ( card `  ( card `  T
) )  =  (
card `  T )
)  ->  ( card `  T )  =  (
aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4037, 38, 39sylancl 666 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =  ( aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4140fveq2d 5876 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  ( cf `  ( aleph ` 
|^| { x  e.  On  |  ( card `  T
)  C_  ( aleph `  x ) } ) ) )
4240, 41oveq12d 6314 . . . . . 6  |-  ( ( 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 ) } ) ) ) )
4340, 42breq12d 4430 . . . . 5  |-  ( ( 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 ) } ) ) ) ) )
445, 43mpbiri 236 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
45 simp1 1005 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  T  e.  Tarski )
46 simp3 1007 . . . . . . . . . . . . 13  |-  ( ( 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
) ) ) )
47 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( card `  T )  e.  _V
48 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( cf `  ( card `  T
) )  e.  _V
4947, 48elmap 7499 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  <->  x :
( cf `  ( card `  T ) ) --> ( card `  T
) )
50 fssxp 5749 . . . . . . . . . . . . . . 15  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  C_  (
( cf `  ( card `  T ) )  X.  ( card `  T
) ) )
5149, 50sylbi 198 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) ) )
5216ex 435 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( x  e.  ( card `  T
)  ->  x  e.  T ) )
5352ssrdv 3467 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( card `  T
)  C_  T )
54 cfle 8673 . . . . . . . . . . . . . . . . 17  |-  ( cf `  ( card `  T
) )  C_  ( card `  T )
55 sstr 3469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( cf `  ( card `  T ) ) 
C_  ( card `  T
)  /\  ( card `  T )  C_  T
)  ->  ( cf `  ( card `  T
) )  C_  T
)
5654, 55mpan 674 . . . . . . . . . . . . . . . 16  |-  ( (
card `  T )  C_  T  ->  ( cf `  ( card `  T
) )  C_  T
)
57 tskxpss 9186 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  C_  T  /\  ( card `  T
)  C_  T )  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
)
58573exp 1204 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  C_  T  ->  ( ( card `  T
)  C_  T  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
5958com23 81 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( ( card `  T )  C_  T  ->  ( ( cf `  ( card `  T ) ) 
C_  T  ->  (
( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
6056, 59mpdi 43 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( ( card `  T )  C_  T  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) )
6153, 60mpd 15 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) ) 
C_  T )
62 sstr2 3468 . . . . . . . . . . . . . 14  |-  ( x 
C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) )  ->  ( ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T  ->  x  C_  T )
)
6351, 61, 62syl2im 39 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( T  e.  Tarski  ->  x  C_  T ) )
6446, 45, 63sylc 62 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  C_  T )
65 simp2 1006 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  ( cf `  ( card `  T
) )  e.  (
card `  T )
)
66 ffn 5737 . . . . . . . . . . . . . . . . 17  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  Fn  ( cf `  ( card `  T ) ) )
67 fndmeng 7644 . . . . . . . . . . . . . . . . 17  |-  ( ( x  Fn  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  e.  _V )  ->  ( cf `  ( card `  T ) ) 
~~  x )
6866, 48, 67sylancl 666 . . . . . . . . . . . . . . . 16  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~~  x
)
6949, 68sylbi 198 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( cf `  ( card `  T ) ) 
~~  x )
7069ensymd 7618 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  ~~  ( cf `  ( card `  T ) ) )
71 cardsdomelir 8397 . . . . . . . . . . . . . 14  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~<  T )
72 ensdomtr 7705 . . . . . . . . . . . . . 14  |-  ( ( x  ~~  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  ~<  T )  ->  x  ~<  T )
7370, 71, 72syl2an 479 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  x  ~<  T )
7446, 65, 73syl2anc 665 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  ~<  T )
75 tskssel 9171 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
7645, 64, 74, 75syl3anc 1264 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  e.  T )
77763expia 1207 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( x  e.  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ->  x  e.  T )
)
7877ssrdv 3467 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) ) 
C_  T )
79 ssdomg 7613 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T ) )
8079imp 430 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T )  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T )
8178, 80syldan 472 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  T )
8224adantr 466 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  T  ~~  ( card `  T )
)
83 domentr 7626 . . . . . . . 8  |-  ( ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )
)
8481, 82, 83syl2anc 665 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  ( card `  T
) )
85 domnsym 7695 . . . . . . 7  |-  ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) )
8684, 85syl 17 . . . . . 6  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
8786ex 435 . . . . 5  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  e.  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) ) )
8887adantr 466 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) ) )
8944, 88mt2d 120 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  -.  ( cf `  ( card `  T ) )  e.  ( card `  T
) )
90 cfon 8674 . . . . . 6  |-  ( cf `  ( card `  T
) )  e.  On
9190, 10onsseli 5547 . . . . 5  |-  ( ( cf `  ( card `  T ) )  C_  ( card `  T )  <->  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
) )
9254, 91mpbi 211 . . . 4  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9392ori 376 . . 3  |-  ( -.  ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9489, 93syl 17 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  (
card `  T )
)
95 elina 9101 . 2  |-  ( (
card `  T )  e.  Inacc 
<->  ( ( card `  T
)  =/=  (/)  /\  ( cf `  ( card `  T
) )  =  (
card `  T )  /\  A. x  e.  (
card `  T ) ~P x  ~<  ( card `  T ) ) )
963, 94, 31, 95syl3anbrc 1189 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e. 
Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078    C_ wss 3433   (/)c0 3758   ~Pcpw 3976   |^|cint 4249   class class class wbr 4417    |-> cmpt 4475    X. cxp 4843   Oncon0 5433    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   omcom 6697    ^m cmap 7471    ~~ cen 7565    ~<_ cdom 7566    ~< csdm 7567  harchar 8062   cardccrd 8359   alephcale 8360   cfccf 8361   Inacccina 9097   Tarskictsk 9162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-ac2 8882
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-smo 7064  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-oi 8016  df-har 8064  df-r1 8225  df-card 8363  df-aleph 8364  df-cf 8365  df-acn 8366  df-ac 8536  df-ina 9099  df-tsk 9163
This theorem is referenced by:  r1tskina  9196  tskuni  9197  inaprc  9250
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