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Theorem gruina 8698
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 3639 . . . 4  |-  ( U  =/=  (/)  <->  E. x  x  e.  U )
2 0ss 3658 . . . . . . . . . . 11  |-  (/)  C_  x
3 gruss 8676 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  (/)  C_  x
)  ->  (/)  e.  U
)
42, 3mp3an3 1269 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  U
)
5 0elon 4637 . . . . . . . . . 10  |-  (/)  e.  On
64, 5jctir 526 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( (/) 
e.  U  /\  (/)  e.  On ) )
7 elin 3532 . . . . . . . . 9  |-  ( (/)  e.  ( U  i^i  On ) 
<->  ( (/)  e.  U  /\  (/)  e.  On ) )
86, 7sylibr 205 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  ( U  i^i  On ) )
9 gruina.1 . . . . . . . 8  |-  A  =  ( U  i^i  On )
108, 9syl6eleqr 2529 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  A
)
11 ne0i 3636 . . . . . . 7  |-  ( (/)  e.  A  ->  A  =/=  (/) )
1210, 11syl 16 . . . . . 6  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  =/=  (/) )
1312expcom 426 . . . . 5  |-  ( x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1413exlimiv 1645 . . . 4  |-  ( E. x  x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
151, 14sylbi 189 . . 3  |-  ( U  =/=  (/)  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1615impcom 421 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  =/=  (/) )
17 grutr 8673 . . . . . . . 8  |-  ( U  e.  Univ  ->  Tr  U
)
18 tron 4607 . . . . . . . 8  |-  Tr  On
19 trin 4315 . . . . . . . 8  |-  ( ( Tr  U  /\  Tr  On )  ->  Tr  ( U  i^i  On ) )
2017, 18, 19sylancl 645 . . . . . . 7  |-  ( U  e.  Univ  ->  Tr  ( U  i^i  On ) )
21 inss2 3564 . . . . . . . . 9  |-  ( U  i^i  On )  C_  On
22 epweon 4767 . . . . . . . . 9  |-  _E  We  On
23 wess 4572 . . . . . . . . 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 4587 . . . . . . 7  |-  ( Ord  ( U  i^i  On ) 
<->  ( Tr  ( U  i^i  On )  /\  _E  We  ( U  i^i  On ) ) )
2720, 25, 26sylanbrc 647 . . . . . 6  |-  ( U  e.  Univ  ->  Ord  ( U  i^i  On ) )
28 inex1g 4349 . . . . . 6  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e. 
_V )
29 elon2 4595 . . . . . 6  |-  ( ( U  i^i  On )  e.  On  <->  ( Ord  ( U  i^i  On )  /\  ( U  i^i  On )  e.  _V )
)
3027, 28, 29sylanbrc 647 . . . . 5  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e.  On )
319, 30syl5eqel 2522 . . . 4  |-  ( U  e.  Univ  ->  A  e.  On )
3231adantr 453 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  On )
33 eloni 4594 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
34 ordirr 4602 . . . . . . 7  |-  ( Ord 
A  ->  -.  A  e.  A )
3533, 34syl 16 . . . . . 6  |-  ( A  e.  On  ->  -.  A  e.  A )
36 elin 3532 . . . . . . . . 9  |-  ( A  e.  ( U  i^i  On )  <->  ( A  e.  U  /\  A  e.  On ) )
3736biimpri 199 . . . . . . . 8  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  ( U  i^i  On ) )
3837, 9syl6eleqr 2529 . . . . . . 7  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  A )
3938expcom 426 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  U  ->  A  e.  A ) )
4035, 39mtod 171 . . . . 5  |-  ( A  e.  On  ->  -.  A  e.  U )
4132, 40syl 16 . . . 4  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  A  e.  U )
42 inss1 3563 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  On )  C_  U
439, 42eqsstri 3380 . . . . . . . . . . . . . . 15  |-  A  C_  U
4443sseli 3346 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  x  e.  U )
45 vex 2961 . . . . . . . . . . . . . . . . 17  |-  x  e. 
_V
4645pwex 4385 . . . . . . . . . . . . . . . 16  |-  ~P x  e.  _V
4746canth2 7263 . . . . . . . . . . . . . . 15  |-  ~P x  ~<  ~P ~P x
4846pwex 4385 . . . . . . . . . . . . . . . . . 18  |-  ~P ~P x  e.  _V
4948cardid 8427 . . . . . . . . . . . . . . . . 17  |-  ( card `  ~P ~P x ) 
~~  ~P ~P x
5049ensymi 7160 . . . . . . . . . . . . . . . 16  |-  ~P ~P x  ~~  ( card `  ~P ~P x )
5131adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  e.  On )
52 grupw 8675 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
53 grupw 8675 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P x  e.  U )  ->  ~P ~P x  e.  U )
5452, 53syldan 458 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  e.  U
)
5531adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  A  e.  On )
56 endom 7137 . . . . . . . . . . . . . . . . . . . . . 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 7836 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( card `  ~P ~P x )  e.  On
59 grudomon 8697 . . . . . . . . . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U  e.  Univ  /\  ( ~P ~P x  e.  U  /\  ( card `  ~P ~P x )  ~<_  ~P ~P x ) )  -> 
( card `  ~P ~P x
)  e.  U )
6157, 60mpanr2 667 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  U )
62 elin 3532 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  e.  ( U  i^i  On )  <->  ( ( card `  ~P ~P x
)  e.  U  /\  ( card `  ~P ~P x
)  e.  On ) )
6362biimpri 199 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  ( U  i^i  On ) )
6463, 9syl6eleqr 2529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  A
)
6561, 58, 64sylancl 645 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  A )
66 onelss 4626 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  e.  A  -> 
( card `  ~P ~P x
)  C_  A )
)
6755, 65, 66sylc 59 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x ) 
C_  A )
6854, 67syldan 458 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  C_  A )
69 ssdomg 7156 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  C_  A  ->  (
card `  ~P ~P x
)  ~<_  A ) )
7051, 68, 69sylc 59 . . . . . . . . . . . . . . . 16  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  ~<_  A )
71 endomtr 7168 . . . . . . . . . . . . . . . 16  |-  ( ( ~P ~P x  ~~  ( card `  ~P ~P x
)  /\  ( card `  ~P ~P x )  ~<_  A )  ->  ~P ~P x  ~<_  A )
7250, 70, 71sylancr 646 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  ~<_  A )
73 sdomdomtr 7243 . . . . . . . . . . . . . . 15  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  A )  ->  ~P x  ~<  A )
7447, 72, 73sylancr 646 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  ~<  A )
7544, 74sylan2 462 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  ~P x  ~<  A )
7675ralrimiva 2791 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  ~P x  ~<  A )
77 inawinalem 8569 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
7831, 76, 77sylc 59 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
7978adantr 453 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
80 winainflem 8573 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A
)
8116, 32, 79, 80syl3anc 1185 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  om  C_  A
)
8245canth2 7263 . . . . . . . . . . . . . 14  |-  x  ~<  ~P x
83 sdomtr 7248 . . . . . . . . . . . . . 14  |-  ( ( x  ~<  ~P x  /\  ~P x  ~<  A )  ->  x  ~<  A )
8482, 75, 83sylancr 646 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  x  ~<  A )
8584ralrimiva 2791 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  x  ~<  A )
86 iscard 7867 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
8731, 85, 86sylanbrc 647 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  ( card `  A )  =  A )
88 cardlim 7864 . . . . . . . . . . . 12  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
89 sseq2 3372 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( om  C_  ( card `  A
)  <->  om  C_  A )
)
90 limeq 4596 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( Lim  ( card `  A
)  <->  Lim  A ) )
9189, 90bibi12d 314 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( ( om  C_  ( card `  A )  <->  Lim  ( card `  A ) )  <->  ( om  C_  A  <->  Lim  A ) ) )
9288, 91mpbii 204 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  <->  Lim  A ) )
9387, 92syl 16 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( om  C_  A  <->  Lim  A ) )
9493adantr 453 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( om  C_  A  <->  Lim  A ) )
9581, 94mpbid 203 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Lim  A )
96 cflm 8135 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  A )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
9732, 95, 96syl2anc 644 . . . . . . 7  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
98 cardon 7836 . . . . . . . . . . . 12  |-  ( card `  y )  e.  On
99 eleq1 2498 . . . . . . . . . . . 12  |-  ( x  =  ( card `  y
)  ->  ( x  e.  On  <->  ( card `  y
)  e.  On ) )
10098, 99mpbiri 226 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  x  e.  On )
101100adantr 453 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  x  e.  On )
102101exlimiv 1645 . . . . . . . . 9  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  ->  x  e.  On )
103102abssi 3420 . . . . . . . 8  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On
104 fvex 5745 . . . . . . . . . 10  |-  ( cf `  A )  e.  _V
10597, 104syl6eqelr 2527 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
106 intex 4359 . . . . . . . . 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 205 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  =/=  (/) )
108 onint 4778 . . . . . . . 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 646 . . . . . . 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 2444 . . . . . . . . 9  |-  ( x  =  ( cf `  A
)  ->  ( x  =  ( card `  y
)  <->  ( cf `  A
)  =  ( card `  y ) ) )
112111anbi1d 687 . . . . . . . 8  |-  ( x  =  ( cf `  A
)  ->  ( (
x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  <->  ( ( cf `  A )  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) ) )
113112exbidv 1637 . . . . . . 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 3084 . . . . . 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 190 . . . . 5  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  E. y
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) ) )
116 simp2rr 1028 . . . . . . . 8  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  =  U. y )
117 simp1l 982 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  U  e.  Univ )
118 simp2rl 1027 . . . . . . . . . . 11  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  A )
119118, 43syl6ss 3362 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  U )
12043sseli 3346 . . . . . . . . . . 11  |-  ( ( cf `  A )  e.  A  ->  ( cf `  A )  e.  U )
1211203ad2ant3 981 . . . . . . . . . 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 984 . . . . . . . . . . 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 2961 . . . . . . . . . . . 12  |-  y  e. 
_V
124123cardid 8427 . . . . . . . . . . 11  |-  ( card `  y )  ~~  y
125122, 124syl6eqbr 4252 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  ~~  y )
126 gruen 8692 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  y  C_  U  /\  ( ( cf `  A )  e.  U  /\  ( cf `  A )  ~~  y ) )  -> 
y  e.  U )
127117, 119, 121, 125, 126syl112anc 1189 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  e.  U )
128 gruuni 8680 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  y  e.  U )  ->  U. y  e.  U )
129117, 127, 128syl2anc 644 . . . . . . . 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 1153 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  (
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  (
( cf `  A
)  e.  A  ->  A  e.  U )
) )
132131exlimdv 1647 . . . . 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 171 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  ( cf `  A )  e.  A )
135 cfon 8140 . . . . 5  |-  ( cf `  A )  e.  On
136 cfle 8139 . . . . . 6  |-  ( cf `  A )  C_  A
137 onsseleq 4625 . . . . . 6  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  C_  A  <->  ( ( cf `  A )  e.  A  \/  ( cf `  A )  =  A ) ) )
138136, 137mpbii 204 . . . . 5  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
139135, 138mpan 653 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
140139ord 368 . . 3  |-  ( A  e.  On  ->  ( -.  ( cf `  A
)  e.  A  -> 
( cf `  A
)  =  A ) )
14132, 134, 140sylc 59 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  =  A )
14276adantr 453 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  ~P x  ~<  A )
143 elina 8567 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
14416, 141, 142, 143syl3anbrc 1139 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   |^|cint 4052   class class class wbr 4215   Tr wtr 4305    _E cep 4495    We wwe 4543   Ord word 4583   Oncon0 4584   Lim wlim 4585   omcom 4848   ` cfv 5457    ~~ cen 7109    ~<_ cdom 7110    ~< csdm 7111   cardccrd 7827   cfccf 7829   Inacccina 8563   Univcgru 8670
This theorem is referenced by:  grur1a  8699  grur1  8700  grutsk  8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-ac2 8348
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-recs 6636  df-1o 6727  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-card 7831  df-cf 7833  df-ac 8002  df-ina 8565  df-gru 8671
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