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Theorem tskcard 9232
Description: An even more direct relationship than r1tskina 9233 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 9003 . . . 4  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =  (/)  <->  T  =  (/) ) )
21necon3bid 2680 . . 3  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =/=  (/)  <->  T  =/=  (/) ) )
32biimpar 492 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =/=  (/) )
4 eqid 2462 . . . . . 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 9034 . . . . 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 3060 . . . . . . . . . . . . 13  |-  x  e. 
_V
76pwex 4600 . . . . . . . . . . . 12  |-  ~P x  e.  _V
87canth2 7751 . . . . . . . . . . 11  |-  ~P x  ~<  ~P ~P x
9 simpl 463 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  e.  Tarski )
10 cardon 8404 . . . . . . . . . . . . . . . . 17  |-  ( card `  T )  e.  On
1110oneli 5549 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  e.  On )
1211adantl 472 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  On )
13 cardsdomelir 8433 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  ~<  T )
1413adantl 472 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  ~<  T )
15 tskord 9231 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  On  /\  x  ~<  T )  ->  x  e.  T )
169, 12, 14, 15syl3anc 1276 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  T )
17 tskpw 9204 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
18 tskpwss 9203 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  ~P x  e.  T )  ->  ~P ~P x  C_  T )
1917, 18syldan 477 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P ~P x  C_  T )
2016, 19syldan 477 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  C_  T )
21 ssdomg 7641 . . . . . . . . . . . . 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 8999 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( card `  T
)  ~~  T )
2423ensymd 7646 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  T  ~~  ( card `  T ) )
2524adantr 471 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  ~~  ( card `  T
) )
26 domentr 7654 . . . . . . . . . . . 12  |-  ( ( ~P ~P x  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ~P ~P x  ~<_  ( card `  T )
)
2722, 25, 26syl2anc 671 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  ( card `  T ) )
28 sdomdomtr 7731 . . . . . . . . . . 11  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  ( card `  T )
)  ->  ~P x  ~<  ( card `  T
) )
298, 27, 28sylancr 674 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P x  ~<  ( card `  T
) )
3029ralrimiva 2814 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  A. x  e.  (
card `  T ) ~P x  ~<  ( card `  T ) )
3130adantr 471 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  A. x  e.  ( card `  T
) ~P x  ~<  (
card `  T )
)
32 inawinalem 9140 . . . . . . . . . 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 9144 . . . . . . . . . 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 1361 . . . . . . . . 9  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) E. y  e.  ( card `  T
) x  ~<  y
)  ->  om  C_  ( card `  T ) )
3633, 35sylan2 481 . . . . . . . 8  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
) )  ->  om  C_  ( card `  T ) )
373, 31, 36syl2anc 671 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om  C_  ( card `  T ) )
38 cardidm 8419 . . . . . . 7  |-  ( card `  ( card `  T
) )  =  (
card `  T )
39 cardaleph 8546 . . . . . . 7  |-  ( ( om  C_  ( card `  T )  /\  ( card `  ( card `  T
) )  =  (
card `  T )
)  ->  ( card `  T )  =  (
aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4037, 38, 39sylancl 673 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =  ( aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4140fveq2d 5892 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  ( cf `  ( aleph ` 
|^| { x  e.  On  |  ( card `  T
)  C_  ( aleph `  x ) } ) ) )
4240, 41oveq12d 6333 . . . . . 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 4429 . . . . 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 241 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
45 simp1 1014 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  T  e.  Tarski )
46 simp3 1016 . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . . 16  |-  ( card `  T )  e.  _V
48 fvex 5898 . . . . . . . . . . . . . . . 16  |-  ( cf `  ( card `  T
) )  e.  _V
4947, 48elmap 7526 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  <->  x :
( cf `  ( card `  T ) ) --> ( card `  T
) )
50 fssxp 5764 . . . . . . . . . . . . . . 15  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  C_  (
( cf `  ( card `  T ) )  X.  ( card `  T
) ) )
5149, 50sylbi 200 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) ) )
5216ex 440 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( x  e.  ( card `  T
)  ->  x  e.  T ) )
5352ssrdv 3450 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( card `  T
)  C_  T )
54 cfle 8710 . . . . . . . . . . . . . . . . 17  |-  ( cf `  ( card `  T
) )  C_  ( card `  T )
55 sstr 3452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( cf `  ( card `  T ) ) 
C_  ( card `  T
)  /\  ( card `  T )  C_  T
)  ->  ( cf `  ( card `  T
) )  C_  T
)
5654, 55mpan 681 . . . . . . . . . . . . . . . 16  |-  ( (
card `  T )  C_  T  ->  ( cf `  ( card `  T
) )  C_  T
)
57 tskxpss 9223 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  C_  T  /\  ( card `  T
)  C_  T )  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
)
58573exp 1214 . . . . . . . . . . . . . . . . 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 3451 . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . 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 5751 . . . . . . . . . . . . . . . . 17  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  Fn  ( cf `  ( card `  T ) ) )
67 fndmeng 7672 . . . . . . . . . . . . . . . . 17  |-  ( ( x  Fn  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  e.  _V )  ->  ( cf `  ( card `  T ) ) 
~~  x )
6866, 48, 67sylancl 673 . . . . . . . . . . . . . . . 16  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~~  x
)
6949, 68sylbi 200 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( cf `  ( card `  T ) ) 
~~  x )
7069ensymd 7646 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  ~~  ( cf `  ( card `  T ) ) )
71 cardsdomelir 8433 . . . . . . . . . . . . . 14  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~<  T )
72 ensdomtr 7734 . . . . . . . . . . . . . 14  |-  ( ( x  ~~  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  ~<  T )  ->  x  ~<  T )
7370, 71, 72syl2an 484 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  x  ~<  T )
7446, 65, 73syl2anc 671 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  ~<  T )
75 tskssel 9208 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
7645, 64, 74, 75syl3anc 1276 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  e.  T )
77763expia 1217 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( x  e.  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ->  x  e.  T )
)
7877ssrdv 3450 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) ) 
C_  T )
79 ssdomg 7641 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T ) )
8079imp 435 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T )  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T )
8178, 80syldan 477 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  T )
8224adantr 471 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  T  ~~  ( card `  T )
)
83 domentr 7654 . . . . . . . 8  |-  ( ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )
)
8481, 82, 83syl2anc 671 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  ( card `  T
) )
85 domnsym 7724 . . . . . . 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 440 . . . . 5  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  e.  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) ) )
8887adantr 471 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) ) )
8944, 88mt2d 122 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  -.  ( cf `  ( card `  T ) )  e.  ( card `  T
) )
90 cfon 8711 . . . . . 6  |-  ( cf `  ( card `  T
) )  e.  On
9190, 10onsseli 5556 . . . . 5  |-  ( ( cf `  ( card `  T ) )  C_  ( card `  T )  <->  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
) )
9254, 91mpbi 213 . . . 4  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9392ori 381 . . 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 9138 . 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 1198 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e. 
Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753   _Vcvv 3057    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   |^|cint 4248   class class class wbr 4416    |-> cmpt 4475    X. cxp 4851   Oncon0 5442    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   omcom 6719    ^m cmap 7498    ~~ cen 7592    ~<_ cdom 7593    ~< csdm 7594  harchar 8097   cardccrd 8395   alephcale 8396   cfccf 8397   Inacccina 9134   Tarskictsk 9199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-ac2 8919
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-smo 7091  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-oi 8051  df-har 8099  df-r1 8261  df-card 8399  df-aleph 8400  df-cf 8401  df-acn 8402  df-ac 8573  df-ina 9136  df-tsk 9200
This theorem is referenced by:  r1tskina  9233  tskuni  9234  inaprc  9287
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