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Theorem tskcard 9159
Description: An even more direct relationship than r1tskina 9160 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 8927 . . . 4  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =  (/)  <->  T  =  (/) ) )
21necon3bid 2725 . . 3  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =/=  (/)  <->  T  =/=  (/) ) )
32biimpar 485 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =/=  (/) )
4 eqid 2467 . . . . . 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 8958 . . . . 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 3116 . . . . . . . . . . . . 13  |-  x  e. 
_V
76pwex 4630 . . . . . . . . . . . 12  |-  ~P x  e.  _V
87canth2 7670 . . . . . . . . . . 11  |-  ~P x  ~<  ~P ~P x
9 simpl 457 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  e.  Tarski )
10 cardon 8325 . . . . . . . . . . . . . . . . 17  |-  ( card `  T )  e.  On
1110oneli 4985 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  e.  On )
1211adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  On )
13 cardsdomelir 8354 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  ~<  T )
1413adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  ~<  T )
15 tskord 9158 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  On  /\  x  ~<  T )  ->  x  e.  T )
169, 12, 14, 15syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  T )
17 tskpw 9131 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
18 tskpwss 9130 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  ~P x  e.  T )  ->  ~P ~P x  C_  T )
1917, 18syldan 470 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P ~P x  C_  T )
2016, 19syldan 470 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  C_  T )
21 ssdomg 7561 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  ( ~P ~P x  C_  T  ->  ~P ~P x  ~<_  T )
)
229, 20, 21sylc 60 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  T )
23 cardidg 8923 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( card `  T
)  ~~  T )
2423ensymd 7566 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  T  ~~  ( card `  T ) )
2524adantr 465 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  ~~  ( card `  T
) )
26 domentr 7574 . . . . . . . . . . . 12  |-  ( ( ~P ~P x  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ~P ~P x  ~<_  ( card `  T )
)
2722, 25, 26syl2anc 661 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  ( card `  T ) )
28 sdomdomtr 7650 . . . . . . . . . . 11  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  ( card `  T )
)  ->  ~P x  ~<  ( card `  T
) )
298, 27, 28sylancr 663 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P x  ~<  ( card `  T
) )
3029ralrimiva 2878 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  A. x  e.  (
card `  T ) ~P x  ~<  ( card `  T ) )
3130adantr 465 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  A. x  e.  ( card `  T
) ~P x  ~<  (
card `  T )
)
32 inawinalem 9067 . . . . . . . . . 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 9071 . . . . . . . . . 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 1312 . . . . . . . . 9  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) E. y  e.  ( card `  T
) x  ~<  y
)  ->  om  C_  ( card `  T ) )
3633, 35sylan2 474 . . . . . . . 8  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
) )  ->  om  C_  ( card `  T ) )
373, 31, 36syl2anc 661 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om  C_  ( card `  T ) )
38 cardidm 8340 . . . . . . 7  |-  ( card `  ( card `  T
) )  =  (
card `  T )
39 cardaleph 8470 . . . . . . 7  |-  ( ( om  C_  ( card `  T )  /\  ( card `  ( card `  T
) )  =  (
card `  T )
)  ->  ( card `  T )  =  (
aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4037, 38, 39sylancl 662 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =  ( aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4140fveq2d 5870 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  ( cf `  ( aleph ` 
|^| { x  e.  On  |  ( card `  T
)  C_  ( aleph `  x ) } ) ) )
4240, 41oveq12d 6302 . . . . . 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 4460 . . . . 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 233 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
45 simp1 996 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  T  e.  Tarski )
46 simp3 998 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . . 16  |-  ( card `  T )  e.  _V
48 fvex 5876 . . . . . . . . . . . . . . . 16  |-  ( cf `  ( card `  T
) )  e.  _V
4947, 48elmap 7447 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  <->  x :
( cf `  ( card `  T ) ) --> ( card `  T
) )
50 fssxp 5743 . . . . . . . . . . . . . . 15  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  C_  (
( cf `  ( card `  T ) )  X.  ( card `  T
) ) )
5149, 50sylbi 195 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) ) )
5216ex 434 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( x  e.  ( card `  T
)  ->  x  e.  T ) )
5352ssrdv 3510 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( card `  T
)  C_  T )
54 cfle 8634 . . . . . . . . . . . . . . . . 17  |-  ( cf `  ( card `  T
) )  C_  ( card `  T )
55 sstr 3512 . . . . . . . . . . . . . . . . 17  |-  ( ( ( cf `  ( card `  T ) ) 
C_  ( card `  T
)  /\  ( card `  T )  C_  T
)  ->  ( cf `  ( card `  T
) )  C_  T
)
5654, 55mpan 670 . . . . . . . . . . . . . . . 16  |-  ( (
card `  T )  C_  T  ->  ( cf `  ( card `  T
) )  C_  T
)
57 tskxpss 9150 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  C_  T  /\  ( card `  T
)  C_  T )  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
)
58573exp 1195 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  C_  T  ->  ( ( card `  T
)  C_  T  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
5958com23 78 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( ( card `  T )  C_  T  ->  ( ( cf `  ( card `  T ) ) 
C_  T  ->  (
( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
6056, 59mpdi 42 . . . . . . . . . . . . . . 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 3511 . . . . . . . . . . . . . 14  |-  ( x 
C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) )  ->  ( ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T  ->  x  C_  T )
)
6351, 61, 62syl2im 38 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( T  e.  Tarski  ->  x  C_  T ) )
6446, 45, 63sylc 60 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  C_  T )
65 simp2 997 . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . . . . 17  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  Fn  ( cf `  ( card `  T ) ) )
67 fndmeng 7592 . . . . . . . . . . . . . . . . 17  |-  ( ( x  Fn  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  e.  _V )  ->  ( cf `  ( card `  T ) ) 
~~  x )
6866, 48, 67sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~~  x
)
6949, 68sylbi 195 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( cf `  ( card `  T ) ) 
~~  x )
7069ensymd 7566 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  ~~  ( cf `  ( card `  T ) ) )
71 cardsdomelir 8354 . . . . . . . . . . . . . 14  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~<  T )
72 ensdomtr 7653 . . . . . . . . . . . . . 14  |-  ( ( x  ~~  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  ~<  T )  ->  x  ~<  T )
7370, 71, 72syl2an 477 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  x  ~<  T )
7446, 65, 73syl2anc 661 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  ~<  T )
75 tskssel 9135 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
7645, 64, 74, 75syl3anc 1228 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  e.  T )
77763expia 1198 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( x  e.  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ->  x  e.  T )
)
7877ssrdv 3510 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) ) 
C_  T )
79 ssdomg 7561 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T ) )
8079imp 429 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T )  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T )
8178, 80syldan 470 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  T )
8224adantr 465 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  T  ~~  ( card `  T )
)
83 domentr 7574 . . . . . . . 8  |-  ( ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )
)
8481, 82, 83syl2anc 661 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  ( card `  T
) )
85 domnsym 7643 . . . . . . 7  |-  ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) )
8684, 85syl 16 . . . . . 6  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
8786ex 434 . . . . 5  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  e.  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) ) )
8887adantr 465 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) ) )
8944, 88mt2d 117 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  -.  ( cf `  ( card `  T ) )  e.  ( card `  T
) )
90 cfon 8635 . . . . . 6  |-  ( cf `  ( card `  T
) )  e.  On
9190, 10onsseli 4992 . . . . 5  |-  ( ( cf `  ( card `  T ) )  C_  ( card `  T )  <->  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
) )
9254, 91mpbi 208 . . . 4  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9392ori 375 . . 3  |-  ( -.  ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9489, 93syl 16 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  (
card `  T )
)
95 elina 9065 . 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 1180 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e. 
Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878    X. cxp 4997    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   omcom 6684    ^m cmap 7420    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515  harchar 7982   cardccrd 8316   alephcale 8317   cfccf 8318   Inacccina 9061   Tarskictsk 9126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-ac2 8843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-smo 7017  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-har 7984  df-r1 8182  df-card 8320  df-aleph 8321  df-cf 8322  df-acn 8323  df-ac 8497  df-ina 9063  df-tsk 9127
This theorem is referenced by:  r1tskina  9160  tskuni  9161  inaprc  9214
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