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Theorem tskcard 8953
Description: An even more direct relationship than r1tskina 8954 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 8721 . . . 4  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =  (/)  <->  T  =  (/) ) )
21necon3bid 2648 . . 3  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =/=  (/)  <->  T  =/=  (/) ) )
32biimpar 485 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =/=  (/) )
4 eqid 2443 . . . . . 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 8752 . . . . 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 2980 . . . . . . . . . . . . 13  |-  x  e. 
_V
76pwex 4480 . . . . . . . . . . . 12  |-  ~P x  e.  _V
87canth2 7469 . . . . . . . . . . 11  |-  ~P x  ~<  ~P ~P x
9 simpl 457 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  e.  Tarski )
10 cardon 8119 . . . . . . . . . . . . . . . . 17  |-  ( card `  T )  e.  On
1110oneli 4831 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  e.  On )
1211adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  On )
13 cardsdomelir 8148 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  ~<  T )
1413adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  ~<  T )
15 tskord 8952 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  On  /\  x  ~<  T )  ->  x  e.  T )
169, 12, 14, 15syl3anc 1218 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  T )
17 tskpw 8925 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
18 tskpwss 8924 . . . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . 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 8717 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( card `  T
)  ~~  T )
2423ensymd 7365 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  T  ~~  ( card `  T ) )
2524adantr 465 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  ~~  ( card `  T
) )
26 domentr 7373 . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 2804 . . . . . . . . 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 8861 . . . . . . . . . 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 8865 . . . . . . . . . 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 1302 . . . . . . . . 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 8134 . . . . . . 7  |-  ( card `  ( card `  T
) )  =  (
card `  T )
39 cardaleph 8264 . . . . . . 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 5700 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  ( cf `  ( aleph ` 
|^| { x  e.  On  |  ( card `  T
)  C_  ( aleph `  x ) } ) ) )
4240, 41oveq12d 6114 . . . . . 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 4310 . . . . 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 988 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  T  e.  Tarski )
46 simp3 990 . . . . . . . . . . . . 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 5706 . . . . . . . . . . . . . . . 16  |-  ( card `  T )  e.  _V
48 fvex 5706 . . . . . . . . . . . . . . . 16  |-  ( cf `  ( card `  T
) )  e.  _V
4947, 48elmap 7246 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  <->  x :
( cf `  ( card `  T ) ) --> ( card `  T
) )
50 fssxp 5575 . . . . . . . . . . . . . . 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 3367 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( card `  T
)  C_  T )
54 cfle 8428 . . . . . . . . . . . . . . . . 17  |-  ( cf `  ( card `  T
) )  C_  ( card `  T )
55 sstr 3369 . . . . . . . . . . . . . . . . 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 8944 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  C_  T  /\  ( card `  T
)  C_  T )  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
)
58573exp 1186 . . . . . . . . . . . . . . . . 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 3368 . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . 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 5564 . . . . . . . . . . . . . . . . 17  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  Fn  ( cf `  ( card `  T ) ) )
67 fndmeng 7391 . . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  ~~  ( cf `  ( card `  T ) ) )
71 cardsdomelir 8148 . . . . . . . . . . . . . 14  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~<  T )
72 ensdomtr 7452 . . . . . . . . . . . . . 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 8929 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
7645, 64, 74, 75syl3anc 1218 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  e.  T )
77763expia 1189 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( x  e.  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ->  x  e.  T )
)
7877ssrdv 3367 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) ) 
C_  T )
79 ssdomg 7360 . . . . . . . . . 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 7373 . . . . . . . 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 7442 . . . . . . 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 8429 . . . . . 6  |-  ( cf `  ( card `  T
) )  e.  On
9190, 10onsseli 4838 . . . . 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 8859 . 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 1172 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   |^|cint 4133   class class class wbr 4297    e. cmpt 4355   Oncon0 4724    X. cxp 4843    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   omcom 6481    ^m cmap 7219    ~~ cen 7312    ~<_ cdom 7313    ~< csdm 7314  harchar 7776   cardccrd 8110   alephcale 8111   cfccf 8112   Inacccina 8855   Tarskictsk 8920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-ac2 8637
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-smo 6812  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-har 7778  df-r1 7976  df-card 8114  df-aleph 8115  df-cf 8116  df-acn 8117  df-ac 8291  df-ina 8857  df-tsk 8921
This theorem is referenced by:  r1tskina  8954  tskuni  8955  inaprc  9008
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