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Theorem tskcard 8656
Description: An even more direct relationship than r1tskina 8657 to get an inacessible cardinal out of a Tarski's class: the size of any nonempty Tarski's 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 8427 . . . 4  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =  (/)  <->  T  =  (/) ) )
21necon3bid 2636 . . 3  |-  ( T  e.  Tarski  ->  ( ( card `  T )  =/=  (/)  <->  T  =/=  (/) ) )
32biimpar 472 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =/=  (/) )
4 eqid 2436 . . . . . 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 8458 . . . . 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 2959 . . . . . . . . . . . . 13  |-  x  e. 
_V
76pwex 4382 . . . . . . . . . . . 12  |-  ~P x  e.  _V
87canth2 7260 . . . . . . . . . . 11  |-  ~P x  ~<  ~P ~P x
9 simpl 444 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  e.  Tarski )
10 cardon 7831 . . . . . . . . . . . . . . . . 17  |-  ( card `  T )  e.  On
1110oneli 4689 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  e.  On )
1211adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  On )
13 cardsdomelir 7860 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( card `  T
)  ->  x  ~<  T )
1413adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  ~<  T )
15 tskord 8655 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  On  /\  x  ~<  T )  ->  x  e.  T )
169, 12, 14, 15syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  x  e.  T )
17 tskpw 8628 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P x  e.  T )
18 tskpwss 8627 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  ~P x  e.  T )  ->  ~P ~P x  C_  T )
1917, 18syldan 457 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  ~P ~P x  C_  T )
2016, 19syldan 457 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  C_  T )
21 ssdomg 7153 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  ( ~P ~P x  C_  T  ->  ~P ~P x  ~<_  T )
)
229, 20, 21sylc 58 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  T )
23 cardidg 8423 . . . . . . . . . . . . . 14  |-  ( T  e.  Tarski  ->  ( card `  T
)  ~~  T )
2423ensymd 7158 . . . . . . . . . . . . 13  |-  ( T  e.  Tarski  ->  T  ~~  ( card `  T ) )
2524adantr 452 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  T  ~~  ( card `  T
) )
26 domentr 7166 . . . . . . . . . . . 12  |-  ( ( ~P ~P x  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ~P ~P x  ~<_  ( card `  T )
)
2722, 25, 26syl2anc 643 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P ~P x  ~<_  ( card `  T ) )
28 sdomdomtr 7240 . . . . . . . . . . 11  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  ( card `  T )
)  ->  ~P x  ~<  ( card `  T
) )
298, 27, 28sylancr 645 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  x  e.  ( card `  T
) )  ->  ~P x  ~<  ( card `  T
) )
3029ralrimiva 2789 . . . . . . . . 9  |-  ( T  e.  Tarski  ->  A. x  e.  (
card `  T ) ~P x  ~<  ( card `  T ) )
3130adantr 452 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  A. x  e.  ( card `  T
) ~P x  ~<  (
card `  T )
)
32 inawinalem 8564 . . . . . . . . . 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 8 . . . . . . . . 9  |-  ( A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
)  ->  A. x  e.  ( card `  T
) E. y  e.  ( card `  T
) x  ~<  y
)
34 winainflem 8568 . . . . . . . . . 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 1267 . . . . . . . . 9  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) E. y  e.  ( card `  T
) x  ~<  y
)  ->  om  C_  ( card `  T ) )
3633, 35sylan2 461 . . . . . . . 8  |-  ( ( ( card `  T
)  =/=  (/)  /\  A. x  e.  ( card `  T ) ~P x  ~<  ( card `  T
) )  ->  om  C_  ( card `  T ) )
373, 31, 36syl2anc 643 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om  C_  ( card `  T ) )
38 cardidm 7846 . . . . . . 7  |-  ( card `  ( card `  T
) )  =  (
card `  T )
39 cardaleph 7970 . . . . . . 7  |-  ( ( om  C_  ( card `  T )  /\  ( card `  ( card `  T
) )  =  (
card `  T )
)  ->  ( card `  T )  =  (
aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4037, 38, 39sylancl 644 . . . . . 6  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  =  ( aleph `  |^| { x  e.  On  |  ( card `  T )  C_  ( aleph `  x ) } ) )
4140fveq2d 5732 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( cf `  ( card `  T
) )  =  ( cf `  ( aleph ` 
|^| { x  e.  On  |  ( card `  T
)  C_  ( aleph `  x ) } ) ) )
4240, 41oveq12d 6099 . . . . . 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 4225 . . . . 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 225 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) )
45 simp1 957 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  T  e.  Tarski )
46 simp3 959 . . . . . . . . . . . . 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 5742 . . . . . . . . . . . . . . . 16  |-  ( card `  T )  e.  _V
48 fvex 5742 . . . . . . . . . . . . . . . 16  |-  ( cf `  ( card `  T
) )  e.  _V
4947, 48elmap 7042 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  <->  x :
( cf `  ( card `  T ) ) --> ( card `  T
) )
50 fssxp 5602 . . . . . . . . . . . . . . 15  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  C_  (
( cf `  ( card `  T ) )  X.  ( card `  T
) ) )
5149, 50sylbi 188 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) ) )
5216ex 424 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( x  e.  ( card `  T
)  ->  x  e.  T ) )
5352ssrdv 3354 . . . . . . . . . . . . . . 15  |-  ( T  e.  Tarski  ->  ( card `  T
)  C_  T )
54 cfle 8134 . . . . . . . . . . . . . . . . 17  |-  ( cf `  ( card `  T
) )  C_  ( card `  T )
55 sstr 3356 . . . . . . . . . . . . . . . . 17  |-  ( ( ( cf `  ( card `  T ) ) 
C_  ( card `  T
)  /\  ( card `  T )  C_  T
)  ->  ( cf `  ( card `  T
) )  C_  T
)
5654, 55mpan 652 . . . . . . . . . . . . . . . 16  |-  ( (
card `  T )  C_  T  ->  ( cf `  ( card `  T
) )  C_  T
)
57 tskxpss 8647 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  C_  T  /\  ( card `  T
)  C_  T )  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
)
58573exp 1152 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  C_  T  ->  ( ( card `  T
)  C_  T  ->  ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
5958com23 74 . . . . . . . . . . . . . . . 16  |-  ( T  e.  Tarski  ->  ( ( card `  T )  C_  T  ->  ( ( cf `  ( card `  T ) ) 
C_  T  ->  (
( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T
) ) )
6056, 59mpdi 40 . . . . . . . . . . . . . . 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 3355 . . . . . . . . . . . . . 14  |-  ( x 
C_  ( ( cf `  ( card `  T
) )  X.  ( card `  T ) )  ->  ( ( ( cf `  ( card `  T ) )  X.  ( card `  T
) )  C_  T  ->  x  C_  T )
)
6351, 61, 62syl2im 36 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( T  e.  Tarski  ->  x  C_  T ) )
6446, 45, 63sylc 58 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  C_  T )
65 simp2 958 . . . . . . . . . . . . 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 5591 . . . . . . . . . . . . . . . . 17  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  x  Fn  ( cf `  ( card `  T ) ) )
67 fndmeng 7183 . . . . . . . . . . . . . . . . 17  |-  ( ( x  Fn  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  e.  _V )  ->  ( cf `  ( card `  T ) ) 
~~  x )
6866, 48, 67sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( x : ( cf `  ( card `  T ) ) --> ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~~  x
)
6949, 68sylbi 188 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  -> 
( cf `  ( card `  T ) ) 
~~  x )
7069ensymd 7158 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ->  x  ~~  ( cf `  ( card `  T ) ) )
71 cardsdomelir 7860 . . . . . . . . . . . . . 14  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  ( cf `  ( card `  T
) )  ~<  T )
72 ensdomtr 7243 . . . . . . . . . . . . . 14  |-  ( ( x  ~~  ( cf `  ( card `  T
) )  /\  ( cf `  ( card `  T
) )  ~<  T )  ->  x  ~<  T )
7370, 71, 72syl2an 464 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  x  ~<  T )
7446, 65, 73syl2anc 643 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  ~<  T )
75 tskssel 8632 . . . . . . . . . . . 12  |-  ( ( T  e.  Tarski  /\  x  C_  T  /\  x  ~<  T )  ->  x  e.  T )
7645, 64, 74, 75syl3anc 1184 . . . . . . . . . . 11  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )  /\  x  e.  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) ) )  ->  x  e.  T )
77763expia 1155 . . . . . . . . . 10  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( x  e.  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ->  x  e.  T )
)
7877ssrdv 3354 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) ) 
C_  T )
79 ssdomg 7153 . . . . . . . . . 10  |-  ( T  e.  Tarski  ->  ( ( (
card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T ) )
8079imp 419 . . . . . . . . 9  |-  ( ( T  e.  Tarski  /\  (
( card `  T )  ^m  ( cf `  ( card `  T ) ) )  C_  T )  ->  ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T )
8178, 80syldan 457 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  T )
8224adantr 452 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  T  ~~  ( card `  T )
)
83 domentr 7166 . . . . . . . 8  |-  ( ( ( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) )  ~<_  T  /\  T  ~~  ( card `  T ) )  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) )  ~<_  (
card `  T )
)
8481, 82, 83syl2anc 643 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  ( cf `  ( card `  T
) )  e.  (
card `  T )
)  ->  ( ( card `  T )  ^m  ( cf `  ( card `  T ) ) )  ~<_  ( card `  T
) )
85 domnsym 7233 . . . . . . 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 424 . . . . 5  |-  ( T  e.  Tarski  ->  ( ( cf `  ( card `  T
) )  e.  (
card `  T )  ->  -.  ( card `  T
)  ~<  ( ( card `  T )  ^m  ( cf `  ( card `  T
) ) ) ) )
8887adantr 452 . . . 4  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (
( cf `  ( card `  T ) )  e.  ( card `  T
)  ->  -.  ( card `  T )  ~< 
( ( card `  T
)  ^m  ( cf `  ( card `  T
) ) ) ) )
8944, 88mt2d 111 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  -.  ( cf `  ( card `  T ) )  e.  ( card `  T
) )
90 cfon 8135 . . . . . 6  |-  ( cf `  ( card `  T
) )  e.  On
9190, 10onsseli 4696 . . . . 5  |-  ( ( cf `  ( card `  T ) )  C_  ( card `  T )  <->  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
) )
9254, 91mpbi 200 . . . 4  |-  ( ( cf `  ( card `  T ) )  e.  ( card `  T
)  \/  ( cf `  ( card `  T
) )  =  (
card `  T )
)
9392ori 365 . . 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 8562 . 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 1138 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e. 
Inacc )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   |^|cint 4050   class class class wbr 4212    e. cmpt 4266   Oncon0 4581   omcom 4845    X. cxp 4876    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108  harchar 7524   cardccrd 7822   alephcale 7823   cfccf 7824   Inacccina 8558   Tarskictsk 8623
This theorem is referenced by:  r1tskina  8657  tskuni  8658  inaprc  8711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-ac2 8343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-smo 6608  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-r1 7690  df-card 7826  df-aleph 7827  df-cf 7828  df-acn 7829  df-ac 7997  df-ina 8560  df-tsk 8624
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