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Theorem gruina 9256
Description: If a Grothendieck universe  U is nonempty, then the height of the ordinals in  U is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)
Hypothesis
Ref Expression
gruina.1  |-  A  =  ( U  i^i  On )
Assertion
Ref Expression
gruina  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc )

Proof of Theorem gruina
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3777 . . . 4  |-  ( U  =/=  (/)  <->  E. x  x  e.  U )
2 0ss 3799 . . . . . . . . . . 11  |-  (/)  C_  x
3 gruss 9234 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  (/)  C_  x
)  ->  (/)  e.  U
)
42, 3mp3an3 1350 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  U
)
5 0elon 5501 . . . . . . . . . 10  |-  (/)  e.  On
64, 5jctir 541 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( (/) 
e.  U  /\  (/)  e.  On ) )
7 elin 3655 . . . . . . . . 9  |-  ( (/)  e.  ( U  i^i  On ) 
<->  ( (/)  e.  U  /\  (/)  e.  On ) )
86, 7sylibr 216 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  ( U  i^i  On ) )
9 gruina.1 . . . . . . . 8  |-  A  =  ( U  i^i  On )
108, 9syl6eleqr 2523 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  A
)
11 ne0i 3773 . . . . . . 7  |-  ( (/)  e.  A  ->  A  =/=  (/) )
1210, 11syl 17 . . . . . 6  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  =/=  (/) )
1312expcom 437 . . . . 5  |-  ( x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1413exlimiv 1771 . . . 4  |-  ( E. x  x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
151, 14sylbi 199 . . 3  |-  ( U  =/=  (/)  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1615impcom 432 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  =/=  (/) )
17 grutr 9231 . . . . . . . 8  |-  ( U  e.  Univ  ->  Tr  U
)
18 tron 5471 . . . . . . . 8  |-  Tr  On
19 trin 4534 . . . . . . . 8  |-  ( ( Tr  U  /\  Tr  On )  ->  Tr  ( U  i^i  On ) )
2017, 18, 19sylancl 667 . . . . . . 7  |-  ( U  e.  Univ  ->  Tr  ( U  i^i  On ) )
21 inss2 3689 . . . . . . . . 9  |-  ( U  i^i  On )  C_  On
22 epweon 6630 . . . . . . . . 9  |-  _E  We  On
23 wess 4846 . . . . . . . . 9  |-  ( ( U  i^i  On ) 
C_  On  ->  (  _E  We  On  ->  _E  We  ( U  i^i  On ) ) )
2421, 22, 23mp2 9 . . . . . . . 8  |-  _E  We  ( U  i^i  On )
2524a1i 11 . . . . . . 7  |-  ( U  e.  Univ  ->  _E  We  ( U  i^i  On ) )
26 df-ord 5451 . . . . . . 7  |-  ( Ord  ( U  i^i  On ) 
<->  ( Tr  ( U  i^i  On )  /\  _E  We  ( U  i^i  On ) ) )
2720, 25, 26sylanbrc 669 . . . . . 6  |-  ( U  e.  Univ  ->  Ord  ( U  i^i  On ) )
28 inex1g 4573 . . . . . 6  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e. 
_V )
29 elon2 5459 . . . . . 6  |-  ( ( U  i^i  On )  e.  On  <->  ( Ord  ( U  i^i  On )  /\  ( U  i^i  On )  e.  _V )
)
3027, 28, 29sylanbrc 669 . . . . 5  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e.  On )
319, 30syl5eqel 2516 . . . 4  |-  ( U  e.  Univ  ->  A  e.  On )
3231adantr 467 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  On )
33 eloni 5458 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
34 ordirr 5466 . . . . . . 7  |-  ( Ord 
A  ->  -.  A  e.  A )
3533, 34syl 17 . . . . . 6  |-  ( A  e.  On  ->  -.  A  e.  A )
36 elin 3655 . . . . . . . . 9  |-  ( A  e.  ( U  i^i  On )  <->  ( A  e.  U  /\  A  e.  On ) )
3736biimpri 210 . . . . . . . 8  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  ( U  i^i  On ) )
3837, 9syl6eleqr 2523 . . . . . . 7  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  A )
3938expcom 437 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  U  ->  A  e.  A ) )
4035, 39mtod 181 . . . . 5  |-  ( A  e.  On  ->  -.  A  e.  U )
4132, 40syl 17 . . . 4  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  A  e.  U )
42 inss1 3688 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  On )  C_  U
439, 42eqsstri 3500 . . . . . . . . . . . . . . 15  |-  A  C_  U
4443sseli 3466 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  x  e.  U )
45 vex 3088 . . . . . . . . . . . . . . . . 17  |-  x  e. 
_V
4645pwex 4613 . . . . . . . . . . . . . . . 16  |-  ~P x  e.  _V
4746canth2 7740 . . . . . . . . . . . . . . 15  |-  ~P x  ~<  ~P ~P x
4846pwex 4613 . . . . . . . . . . . . . . . . . 18  |-  ~P ~P x  e.  _V
4948cardid 8985 . . . . . . . . . . . . . . . . 17  |-  ( card `  ~P ~P x ) 
~~  ~P ~P x
5049ensymi 7635 . . . . . . . . . . . . . . . 16  |-  ~P ~P x  ~~  ( card `  ~P ~P x )
5131adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  e.  On )
52 grupw 9233 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
53 grupw 9233 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P x  e.  U )  ->  ~P ~P x  e.  U )
5452, 53syldan 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  e.  U
)
5531adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  A  e.  On )
56 endom 7612 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  ~~  ~P ~P x  ->  ( card `  ~P ~P x )  ~<_  ~P ~P x )
5749, 56ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  ( card `  ~P ~P x )  ~<_  ~P ~P x
58 cardon 8392 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( card `  ~P ~P x )  e.  On
59 grudomon 9255 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( U  e.  Univ  /\  ( card `  ~P ~P x
)  e.  On  /\  ( ~P ~P x  e.  U  /\  ( card `  ~P ~P x )  ~<_  ~P ~P x ) )  ->  ( card `  ~P ~P x )  e.  U )
6058, 59mp3an2 1349 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U  e.  Univ  /\  ( ~P ~P x  e.  U  /\  ( card `  ~P ~P x )  ~<_  ~P ~P x ) )  -> 
( card `  ~P ~P x
)  e.  U )
6157, 60mpanr2 689 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  U )
62 elin 3655 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  e.  ( U  i^i  On )  <->  ( ( card `  ~P ~P x
)  e.  U  /\  ( card `  ~P ~P x
)  e.  On ) )
6362biimpri 210 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  ( U  i^i  On ) )
6463, 9syl6eleqr 2523 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  A
)
6561, 58, 64sylancl 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  A )
66 onelss 5490 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  e.  A  -> 
( card `  ~P ~P x
)  C_  A )
)
6755, 65, 66sylc 63 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x ) 
C_  A )
6854, 67syldan 473 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  C_  A )
69 ssdomg 7631 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  C_  A  ->  (
card `  ~P ~P x
)  ~<_  A ) )
7051, 68, 69sylc 63 . . . . . . . . . . . . . . . 16  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  ~<_  A )
71 endomtr 7643 . . . . . . . . . . . . . . . 16  |-  ( ( ~P ~P x  ~~  ( card `  ~P ~P x
)  /\  ( card `  ~P ~P x )  ~<_  A )  ->  ~P ~P x  ~<_  A )
7250, 70, 71sylancr 668 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  ~<_  A )
73 sdomdomtr 7720 . . . . . . . . . . . . . . 15  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  A )  ->  ~P x  ~<  A )
7447, 72, 73sylancr 668 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  ~<  A )
7544, 74sylan2 477 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  ~P x  ~<  A )
7675ralrimiva 2841 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  ~P x  ~<  A )
77 inawinalem 9127 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
7831, 76, 77sylc 63 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
7978adantr 467 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
80 winainflem 9131 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A
)
8116, 32, 79, 80syl3anc 1265 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  om  C_  A
)
8245canth2 7740 . . . . . . . . . . . . . 14  |-  x  ~<  ~P x
83 sdomtr 7725 . . . . . . . . . . . . . 14  |-  ( ( x  ~<  ~P x  /\  ~P x  ~<  A )  ->  x  ~<  A )
8482, 75, 83sylancr 668 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  x  ~<  A )
8584ralrimiva 2841 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  x  ~<  A )
86 iscard 8423 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
8731, 85, 86sylanbrc 669 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  ( card `  A )  =  A )
88 cardlim 8420 . . . . . . . . . . . 12  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
89 sseq2 3492 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( om  C_  ( card `  A
)  <->  om  C_  A )
)
90 limeq 5460 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( Lim  ( card `  A
)  <->  Lim  A ) )
9189, 90bibi12d 323 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( ( om  C_  ( card `  A )  <->  Lim  ( card `  A ) )  <->  ( om  C_  A  <->  Lim  A ) ) )
9288, 91mpbii 215 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  <->  Lim  A ) )
9387, 92syl 17 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( om  C_  A  <->  Lim  A ) )
9493adantr 467 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( om  C_  A  <->  Lim  A ) )
9581, 94mpbid 214 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Lim  A )
96 cflm 8693 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  A )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
9732, 95, 96syl2anc 666 . . . . . . 7  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
98 cardon 8392 . . . . . . . . . . . 12  |-  ( card `  y )  e.  On
99 eleq1 2496 . . . . . . . . . . . 12  |-  ( x  =  ( card `  y
)  ->  ( x  e.  On  <->  ( card `  y
)  e.  On ) )
10098, 99mpbiri 237 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  x  e.  On )
101100adantr 467 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  x  e.  On )
102101exlimiv 1771 . . . . . . . . 9  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  ->  x  e.  On )
103102abssi 3542 . . . . . . . 8  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On
104 fvex 5897 . . . . . . . . . 10  |-  ( cf `  A )  e.  _V
10597, 104syl6eqelr 2521 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
106 intex 4586 . . . . . . . . 9  |-  ( { x  |  E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A  =  U. y ) ) }  =/=  (/)  <->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
107105, 106sylibr 216 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  =/=  (/) )
108 onint 6642 . . . . . . . 8  |-  ( ( { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On  /\  {
x  |  E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A  =  U. y ) ) }  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
109103, 107, 108sylancr 668 . . . . . . 7  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
11097, 109eqeltrd 2512 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  e. 
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
111 eqeq1 2427 . . . . . . . . 9  |-  ( x  =  ( cf `  A
)  ->  ( x  =  ( card `  y
)  <->  ( cf `  A
)  =  ( card `  y ) ) )
112111anbi1d 710 . . . . . . . 8  |-  ( x  =  ( cf `  A
)  ->  ( (
x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  <->  ( ( cf `  A )  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) ) )
113112exbidv 1763 . . . . . . 7  |-  ( x  =  ( cf `  A
)  ->  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  <->  E. y ( ( cf `  A )  =  (
card `  y )  /\  ( y  C_  A  /\  A  =  U. y ) ) ) )
114104, 113elab 3223 . . . . . 6  |-  ( ( cf `  A )  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  <->  E. y ( ( cf `  A )  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) )
115110, 114sylib 200 . . . . 5  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  E. y
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) ) )
116 simp2rr 1076 . . . . . . . 8  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  =  U. y )
117 simp1l 1030 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  U  e.  Univ )
118 simp2rl 1075 . . . . . . . . . . 11  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  A )
119118, 43syl6ss 3482 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  U )
12043sseli 3466 . . . . . . . . . . 11  |-  ( ( cf `  A )  e.  A  ->  ( cf `  A )  e.  U )
1211203ad2ant3 1029 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  e.  U )
122 simp2l 1032 . . . . . . . . . . 11  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  =  ( card `  y
) )
123 vex 3088 . . . . . . . . . . . 12  |-  y  e. 
_V
124123cardid 8985 . . . . . . . . . . 11  |-  ( card `  y )  ~~  y
125122, 124syl6eqbr 4467 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  ~~  y )
126 gruen 9250 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  y  C_  U  /\  ( ( cf `  A )  e.  U  /\  ( cf `  A )  ~~  y ) )  -> 
y  e.  U )
127117, 119, 121, 125, 126syl112anc 1269 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  e.  U )
128 gruuni 9238 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  y  e.  U )  ->  U. y  e.  U )
129117, 127, 128syl2anc 666 . . . . . . . 8  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  U. y  e.  U )
130116, 129eqeltrd 2512 . . . . . . 7  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  e.  U )
1311303exp 1205 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  (
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  (
( cf `  A
)  e.  A  ->  A  e.  U )
) )
132131exlimdv 1773 . . . . 5  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( E. y ( ( cf `  A )  =  (
card `  y )  /\  ( y  C_  A  /\  A  =  U. y ) )  -> 
( ( cf `  A
)  e.  A  ->  A  e.  U )
) )
133115, 132mpd 15 . . . 4  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  (
( cf `  A
)  e.  A  ->  A  e.  U )
)
13441, 133mtod 181 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  ( cf `  A )  e.  A )
135 cfon 8698 . . . . 5  |-  ( cf `  A )  e.  On
136 cfle 8697 . . . . . 6  |-  ( cf `  A )  C_  A
137 onsseleq 5489 . . . . . 6  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  C_  A  <->  ( ( cf `  A )  e.  A  \/  ( cf `  A )  =  A ) ) )
138136, 137mpbii 215 . . . . 5  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
139135, 138mpan 675 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
140139ord 379 . . 3  |-  ( A  e.  On  ->  ( -.  ( cf `  A
)  e.  A  -> 
( cf `  A
)  =  A ) )
14132, 134, 140sylc 63 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  =  A )
14276adantr 467 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  ~P x  ~<  A )
143 elina 9125 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
14416, 141, 142, 143syl3anbrc 1190 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 983    = wceq 1438   E.wex 1658    e. wcel 1873   {cab 2408    =/= wne 2619   A.wral 2776   E.wrex 2777   _Vcvv 3085    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3987   U.cuni 4225   |^|cint 4261   class class class wbr 4429   Tr wtr 4524    _E cep 4768    We wwe 4817   Ord word 5447   Oncon0 5448   Lim wlim 5449   ` cfv 5607   omcom 6712    ~~ cen 7583    ~<_ cdom 7584    ~< csdm 7585   cardccrd 8383   cfccf 8385   Inacccina 9121   Univcgru 9228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-ac2 8906
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-se 4819  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-isom 5616  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-om 6713  df-wrecs 7045  df-recs 7107  df-1o 7199  df-er 7380  df-map 7491  df-en 7587  df-dom 7588  df-sdom 7589  df-card 8387  df-cf 8389  df-ac 8560  df-ina 9123  df-gru 9229
This theorem is referenced by:  grur1a  9257  grur1  9258  grutsk  9260
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