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Theorem gruina 8685
Description: If a Grothendieck's 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 3629 . . . 4  |-  ( U  =/=  (/)  <->  E. x  x  e.  U )
2 0ss 3648 . . . . . . . . . . 11  |-  (/)  C_  x
3 gruss 8663 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  (/)  C_  x
)  ->  (/)  e.  U
)
42, 3mp3an3 1268 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  U
)
5 0elon 4626 . . . . . . . . . 10  |-  (/)  e.  On
64, 5jctir 525 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( (/) 
e.  U  /\  (/)  e.  On ) )
7 elin 3522 . . . . . . . . 9  |-  ( (/)  e.  ( U  i^i  On ) 
<->  ( (/)  e.  U  /\  (/)  e.  On ) )
86, 7sylibr 204 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  ( U  i^i  On ) )
9 gruina.1 . . . . . . . 8  |-  A  =  ( U  i^i  On )
108, 9syl6eleqr 2526 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  A
)
11 ne0i 3626 . . . . . . 7  |-  ( (/)  e.  A  ->  A  =/=  (/) )
1210, 11syl 16 . . . . . 6  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  =/=  (/) )
1312expcom 425 . . . . 5  |-  ( x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1413exlimiv 1644 . . . 4  |-  ( E. x  x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
151, 14sylbi 188 . . 3  |-  ( U  =/=  (/)  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1615impcom 420 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  =/=  (/) )
17 grutr 8660 . . . . . . . 8  |-  ( U  e.  Univ  ->  Tr  U
)
18 tron 4596 . . . . . . . 8  |-  Tr  On
19 trin 4304 . . . . . . . 8  |-  ( ( Tr  U  /\  Tr  On )  ->  Tr  ( U  i^i  On ) )
2017, 18, 19sylancl 644 . . . . . . 7  |-  ( U  e.  Univ  ->  Tr  ( U  i^i  On ) )
21 inss2 3554 . . . . . . . . 9  |-  ( U  i^i  On )  C_  On
22 epweon 4756 . . . . . . . . 9  |-  _E  We  On
23 wess 4561 . . . . . . . . 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 4576 . . . . . . 7  |-  ( Ord  ( U  i^i  On ) 
<->  ( Tr  ( U  i^i  On )  /\  _E  We  ( U  i^i  On ) ) )
2720, 25, 26sylanbrc 646 . . . . . 6  |-  ( U  e.  Univ  ->  Ord  ( U  i^i  On ) )
28 inex1g 4338 . . . . . 6  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e. 
_V )
29 elon2 4584 . . . . . 6  |-  ( ( U  i^i  On )  e.  On  <->  ( Ord  ( U  i^i  On )  /\  ( U  i^i  On )  e.  _V )
)
3027, 28, 29sylanbrc 646 . . . . 5  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e.  On )
319, 30syl5eqel 2519 . . . 4  |-  ( U  e.  Univ  ->  A  e.  On )
3231adantr 452 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  On )
33 eloni 4583 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
34 ordirr 4591 . . . . . . 7  |-  ( Ord 
A  ->  -.  A  e.  A )
3533, 34syl 16 . . . . . 6  |-  ( A  e.  On  ->  -.  A  e.  A )
36 elin 3522 . . . . . . . . 9  |-  ( A  e.  ( U  i^i  On )  <->  ( A  e.  U  /\  A  e.  On ) )
3736biimpri 198 . . . . . . . 8  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  ( U  i^i  On ) )
3837, 9syl6eleqr 2526 . . . . . . 7  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  A )
3938expcom 425 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  U  ->  A  e.  A ) )
4035, 39mtod 170 . . . . 5  |-  ( A  e.  On  ->  -.  A  e.  U )
4132, 40syl 16 . . . 4  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  A  e.  U )
42 inss1 3553 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  On )  C_  U
439, 42eqsstri 3370 . . . . . . . . . . . . . . 15  |-  A  C_  U
4443sseli 3336 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  x  e.  U )
45 vex 2951 . . . . . . . . . . . . . . . . 17  |-  x  e. 
_V
4645pwex 4374 . . . . . . . . . . . . . . . 16  |-  ~P x  e.  _V
4746canth2 7252 . . . . . . . . . . . . . . 15  |-  ~P x  ~<  ~P ~P x
4846pwex 4374 . . . . . . . . . . . . . . . . . 18  |-  ~P ~P x  e.  _V
4948cardid 8414 . . . . . . . . . . . . . . . . 17  |-  ( card `  ~P ~P x ) 
~~  ~P ~P x
5049ensymi 7149 . . . . . . . . . . . . . . . 16  |-  ~P ~P x  ~~  ( card `  ~P ~P x )
5131adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  e.  On )
52 grupw 8662 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
53 grupw 8662 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P x  e.  U )  ->  ~P ~P x  e.  U )
5452, 53syldan 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  e.  U
)
5531adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  A  e.  On )
56 endom 7126 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  ~~  ~P ~P x  ->  ( card `  ~P ~P x )  ~<_  ~P ~P x )
5749, 56ax-mp 8 . . . . . . . . . . . . . . . . . . . . 21  |-  ( card `  ~P ~P x )  ~<_  ~P ~P x
58 cardon 7823 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( card `  ~P ~P x )  e.  On
59 grudomon 8684 . . . . . . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U  e.  Univ  /\  ( ~P ~P x  e.  U  /\  ( card `  ~P ~P x )  ~<_  ~P ~P x ) )  -> 
( card `  ~P ~P x
)  e.  U )
6157, 60mpanr2 666 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  U )
62 elin 3522 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  e.  ( U  i^i  On )  <->  ( ( card `  ~P ~P x
)  e.  U  /\  ( card `  ~P ~P x
)  e.  On ) )
6362biimpri 198 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  ( U  i^i  On ) )
6463, 9syl6eleqr 2526 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  A
)
6561, 58, 64sylancl 644 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  A )
66 onelss 4615 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  e.  A  -> 
( card `  ~P ~P x
)  C_  A )
)
6755, 65, 66sylc 58 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x ) 
C_  A )
6854, 67syldan 457 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  C_  A )
69 ssdomg 7145 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  C_  A  ->  (
card `  ~P ~P x
)  ~<_  A ) )
7051, 68, 69sylc 58 . . . . . . . . . . . . . . . 16  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  ~<_  A )
71 endomtr 7157 . . . . . . . . . . . . . . . 16  |-  ( ( ~P ~P x  ~~  ( card `  ~P ~P x
)  /\  ( card `  ~P ~P x )  ~<_  A )  ->  ~P ~P x  ~<_  A )
7250, 70, 71sylancr 645 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  ~<_  A )
73 sdomdomtr 7232 . . . . . . . . . . . . . . 15  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  A )  ->  ~P x  ~<  A )
7447, 72, 73sylancr 645 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  ~<  A )
7544, 74sylan2 461 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  ~P x  ~<  A )
7675ralrimiva 2781 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  ~P x  ~<  A )
77 inawinalem 8556 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
7831, 76, 77sylc 58 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
7978adantr 452 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
80 winainflem 8560 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A
)
8116, 32, 79, 80syl3anc 1184 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  om  C_  A
)
8245canth2 7252 . . . . . . . . . . . . . 14  |-  x  ~<  ~P x
83 sdomtr 7237 . . . . . . . . . . . . . 14  |-  ( ( x  ~<  ~P x  /\  ~P x  ~<  A )  ->  x  ~<  A )
8482, 75, 83sylancr 645 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  x  ~<  A )
8584ralrimiva 2781 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  x  ~<  A )
86 iscard 7854 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
8731, 85, 86sylanbrc 646 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  ( card `  A )  =  A )
88 cardlim 7851 . . . . . . . . . . . 12  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
89 sseq2 3362 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( om  C_  ( card `  A
)  <->  om  C_  A )
)
90 limeq 4585 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( Lim  ( card `  A
)  <->  Lim  A ) )
9189, 90bibi12d 313 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( ( om  C_  ( card `  A )  <->  Lim  ( card `  A ) )  <->  ( om  C_  A  <->  Lim  A ) ) )
9288, 91mpbii 203 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  <->  Lim  A ) )
9387, 92syl 16 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( om  C_  A  <->  Lim  A ) )
9493adantr 452 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( om  C_  A  <->  Lim  A ) )
9581, 94mpbid 202 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Lim  A )
96 cflm 8122 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  A )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
9732, 95, 96syl2anc 643 . . . . . . 7  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
98 cardon 7823 . . . . . . . . . . . 12  |-  ( card `  y )  e.  On
99 eleq1 2495 . . . . . . . . . . . 12  |-  ( x  =  ( card `  y
)  ->  ( x  e.  On  <->  ( card `  y
)  e.  On ) )
10098, 99mpbiri 225 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  x  e.  On )
101100adantr 452 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  x  e.  On )
102101exlimiv 1644 . . . . . . . . 9  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  ->  x  e.  On )
103102abssi 3410 . . . . . . . 8  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On
104 fvex 5734 . . . . . . . . . 10  |-  ( cf `  A )  e.  _V
10597, 104syl6eqelr 2524 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
106 intex 4348 . . . . . . . . 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 204 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  =/=  (/) )
108 onint 4767 . . . . . . . 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 645 . . . . . . 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 2509 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  e. 
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
111 eqeq1 2441 . . . . . . . . 9  |-  ( x  =  ( cf `  A
)  ->  ( x  =  ( card `  y
)  <->  ( cf `  A
)  =  ( card `  y ) ) )
112111anbi1d 686 . . . . . . . 8  |-  ( x  =  ( cf `  A
)  ->  ( (
x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  <->  ( ( cf `  A )  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) ) )
113112exbidv 1636 . . . . . . 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 3074 . . . . . 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 189 . . . . 5  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  E. y
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) ) )
116 simp2rr 1027 . . . . . . . 8  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  =  U. y )
117 simp1l 981 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  U  e.  Univ )
118 simp2rl 1026 . . . . . . . . . . 11  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  A )
119118, 43syl6ss 3352 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  U )
12043sseli 3336 . . . . . . . . . . 11  |-  ( ( cf `  A )  e.  A  ->  ( cf `  A )  e.  U )
1211203ad2ant3 980 . . . . . . . . . 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 983 . . . . . . . . . . 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 2951 . . . . . . . . . . . 12  |-  y  e. 
_V
124123cardid 8414 . . . . . . . . . . 11  |-  ( card `  y )  ~~  y
125122, 124syl6eqbr 4241 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  ~~  y )
126 gruen 8679 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  y  C_  U  /\  ( ( cf `  A )  e.  U  /\  ( cf `  A )  ~~  y ) )  -> 
y  e.  U )
127117, 119, 121, 125, 126syl112anc 1188 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  e.  U )
128 gruuni 8667 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  y  e.  U )  ->  U. y  e.  U )
129117, 127, 128syl2anc 643 . . . . . . . 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 2509 . . . . . . 7  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  e.  U )
1311303exp 1152 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  (
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  (
( cf `  A
)  e.  A  ->  A  e.  U )
) )
132131exlimdv 1646 . . . . 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 170 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  ( cf `  A )  e.  A )
135 cfon 8127 . . . . 5  |-  ( cf `  A )  e.  On
136 cfle 8126 . . . . . 6  |-  ( cf `  A )  C_  A
137 onsseleq 4614 . . . . . 6  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  C_  A  <->  ( ( cf `  A )  e.  A  \/  ( cf `  A )  =  A ) ) )
138136, 137mpbii 203 . . . . 5  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
139135, 138mpan 652 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
140139ord 367 . . 3  |-  ( A  e.  On  ->  ( -.  ( cf `  A
)  e.  A  -> 
( cf `  A
)  =  A ) )
14132, 134, 140sylc 58 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  =  A )
14276adantr 452 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  ~P x  ~<  A )
143 elina 8554 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
14416, 141, 142, 143syl3anbrc 1138 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   class class class wbr 4204   Tr wtr 4294    _E cep 4484    We wwe 4532   Ord word 4572   Oncon0 4573   Lim wlim 4574   omcom 4837   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   cardccrd 7814   cfccf 7816   Inacccina 8550   Univcgru 8657
This theorem is referenced by:  grur1a  8686  grur1  8687  grutsk  8689
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-ac2 8335
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-1o 6716  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-card 7818  df-cf 7820  df-ac 7989  df-ina 8552  df-gru 8658
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