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Theorem gruina 9196
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 3794 . . . 4  |-  ( U  =/=  (/)  <->  E. x  x  e.  U )
2 0ss 3814 . . . . . . . . . . 11  |-  (/)  C_  x
3 gruss 9174 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  (/)  C_  x
)  ->  (/)  e.  U
)
42, 3mp3an3 1313 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  U
)
5 0elon 4931 . . . . . . . . . 10  |-  (/)  e.  On
64, 5jctir 538 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( (/) 
e.  U  /\  (/)  e.  On ) )
7 elin 3687 . . . . . . . . 9  |-  ( (/)  e.  ( U  i^i  On ) 
<->  ( (/)  e.  U  /\  (/)  e.  On ) )
86, 7sylibr 212 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  ( U  i^i  On ) )
9 gruina.1 . . . . . . . 8  |-  A  =  ( U  i^i  On )
108, 9syl6eleqr 2566 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  A
)
11 ne0i 3791 . . . . . . 7  |-  ( (/)  e.  A  ->  A  =/=  (/) )
1210, 11syl 16 . . . . . 6  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  =/=  (/) )
1312expcom 435 . . . . 5  |-  ( x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1413exlimiv 1698 . . . 4  |-  ( E. x  x  e.  U  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
151, 14sylbi 195 . . 3  |-  ( U  =/=  (/)  ->  ( U  e.  Univ  ->  A  =/=  (/) ) )
1615impcom 430 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  =/=  (/) )
17 grutr 9171 . . . . . . . 8  |-  ( U  e.  Univ  ->  Tr  U
)
18 tron 4901 . . . . . . . 8  |-  Tr  On
19 trin 4550 . . . . . . . 8  |-  ( ( Tr  U  /\  Tr  On )  ->  Tr  ( U  i^i  On ) )
2017, 18, 19sylancl 662 . . . . . . 7  |-  ( U  e.  Univ  ->  Tr  ( U  i^i  On ) )
21 inss2 3719 . . . . . . . . 9  |-  ( U  i^i  On )  C_  On
22 epweon 6603 . . . . . . . . 9  |-  _E  We  On
23 wess 4866 . . . . . . . . 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 4881 . . . . . . 7  |-  ( Ord  ( U  i^i  On ) 
<->  ( Tr  ( U  i^i  On )  /\  _E  We  ( U  i^i  On ) ) )
2720, 25, 26sylanbrc 664 . . . . . 6  |-  ( U  e.  Univ  ->  Ord  ( U  i^i  On ) )
28 inex1g 4590 . . . . . 6  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e. 
_V )
29 elon2 4889 . . . . . 6  |-  ( ( U  i^i  On )  e.  On  <->  ( Ord  ( U  i^i  On )  /\  ( U  i^i  On )  e.  _V )
)
3027, 28, 29sylanbrc 664 . . . . 5  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e.  On )
319, 30syl5eqel 2559 . . . 4  |-  ( U  e.  Univ  ->  A  e.  On )
3231adantr 465 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  On )
33 eloni 4888 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
34 ordirr 4896 . . . . . . 7  |-  ( Ord 
A  ->  -.  A  e.  A )
3533, 34syl 16 . . . . . 6  |-  ( A  e.  On  ->  -.  A  e.  A )
36 elin 3687 . . . . . . . . 9  |-  ( A  e.  ( U  i^i  On )  <->  ( A  e.  U  /\  A  e.  On ) )
3736biimpri 206 . . . . . . . 8  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  ( U  i^i  On ) )
3837, 9syl6eleqr 2566 . . . . . . 7  |-  ( ( A  e.  U  /\  A  e.  On )  ->  A  e.  A )
3938expcom 435 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  U  ->  A  e.  A ) )
4035, 39mtod 177 . . . . 5  |-  ( A  e.  On  ->  -.  A  e.  U )
4132, 40syl 16 . . . 4  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  A  e.  U )
42 inss1 3718 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  On )  C_  U
439, 42eqsstri 3534 . . . . . . . . . . . . . . 15  |-  A  C_  U
4443sseli 3500 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  x  e.  U )
45 vex 3116 . . . . . . . . . . . . . . . . 17  |-  x  e. 
_V
4645pwex 4630 . . . . . . . . . . . . . . . 16  |-  ~P x  e.  _V
4746canth2 7670 . . . . . . . . . . . . . . 15  |-  ~P x  ~<  ~P ~P x
4846pwex 4630 . . . . . . . . . . . . . . . . . 18  |-  ~P ~P x  e.  _V
4948cardid 8922 . . . . . . . . . . . . . . . . 17  |-  ( card `  ~P ~P x ) 
~~  ~P ~P x
5049ensymi 7565 . . . . . . . . . . . . . . . 16  |-  ~P ~P x  ~~  ( card `  ~P ~P x )
5131adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  e.  On )
52 grupw 9173 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
53 grupw 9173 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P x  e.  U )  ->  ~P ~P x  e.  U )
5452, 53syldan 470 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  e.  U
)
5531adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  A  e.  On )
56 endom 7542 . . . . . . . . . . . . . . . . . . . . . 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 8325 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( card `  ~P ~P x )  e.  On
59 grudomon 9195 . . . . . . . . . . . . . . . . . . . . . 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 1312 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U  e.  Univ  /\  ( ~P ~P x  e.  U  /\  ( card `  ~P ~P x )  ~<_  ~P ~P x ) )  -> 
( card `  ~P ~P x
)  e.  U )
6157, 60mpanr2 684 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  U )
62 elin 3687 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
card `  ~P ~P x
)  e.  ( U  i^i  On )  <->  ( ( card `  ~P ~P x
)  e.  U  /\  ( card `  ~P ~P x
)  e.  On ) )
6362biimpri 206 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  ( U  i^i  On ) )
6463, 9syl6eleqr 2566 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( card `  ~P ~P x )  e.  U  /\  ( card `  ~P ~P x )  e.  On )  ->  ( card `  ~P ~P x )  e.  A
)
6561, 58, 64sylancl 662 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x )  e.  A )
66 onelss 4920 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  e.  A  -> 
( card `  ~P ~P x
)  C_  A )
)
6755, 65, 66sylc 60 . . . . . . . . . . . . . . . . . 18  |-  ( ( U  e.  Univ  /\  ~P ~P x  e.  U
)  ->  ( card `  ~P ~P x ) 
C_  A )
6854, 67syldan 470 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  C_  A )
69 ssdomg 7561 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  On  ->  (
( card `  ~P ~P x
)  C_  A  ->  (
card `  ~P ~P x
)  ~<_  A ) )
7051, 68, 69sylc 60 . . . . . . . . . . . . . . . 16  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( card `  ~P ~P x
)  ~<_  A )
71 endomtr 7573 . . . . . . . . . . . . . . . 16  |-  ( ( ~P ~P x  ~~  ( card `  ~P ~P x
)  /\  ( card `  ~P ~P x )  ~<_  A )  ->  ~P ~P x  ~<_  A )
7250, 70, 71sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P ~P x  ~<_  A )
73 sdomdomtr 7650 . . . . . . . . . . . . . . 15  |-  ( ( ~P x  ~<  ~P ~P x  /\  ~P ~P x  ~<_  A )  ->  ~P x  ~<  A )
7447, 72, 73sylancr 663 . . . . . . . . . . . . . 14  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  ~<  A )
7544, 74sylan2 474 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  ~P x  ~<  A )
7675ralrimiva 2878 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  ~P x  ~<  A )
77 inawinalem 9067 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
7831, 76, 77sylc 60 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
7978adantr 465 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  E. y  e.  A  x  ~<  y )
80 winainflem 9071 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A
)
8116, 32, 79, 80syl3anc 1228 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  om  C_  A
)
8245canth2 7670 . . . . . . . . . . . . . 14  |-  x  ~<  ~P x
83 sdomtr 7655 . . . . . . . . . . . . . 14  |-  ( ( x  ~<  ~P x  /\  ~P x  ~<  A )  ->  x  ~<  A )
8482, 75, 83sylancr 663 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  x  ~<  A )
8584ralrimiva 2878 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  x  ~<  A )
86 iscard 8356 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
8731, 85, 86sylanbrc 664 . . . . . . . . . . 11  |-  ( U  e.  Univ  ->  ( card `  A )  =  A )
88 cardlim 8353 . . . . . . . . . . . 12  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
89 sseq2 3526 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( om  C_  ( card `  A
)  <->  om  C_  A )
)
90 limeq 4890 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( Lim  ( card `  A
)  <->  Lim  A ) )
9189, 90bibi12d 321 . . . . . . . . . . . 12  |-  ( (
card `  A )  =  A  ->  ( ( om  C_  ( card `  A )  <->  Lim  ( card `  A ) )  <->  ( om  C_  A  <->  Lim  A ) ) )
9288, 91mpbii 211 . . . . . . . . . . 11  |-  ( (
card `  A )  =  A  ->  ( om  C_  A  <->  Lim  A ) )
9387, 92syl 16 . . . . . . . . . 10  |-  ( U  e.  Univ  ->  ( om  C_  A  <->  Lim  A ) )
9493adantr 465 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( om  C_  A  <->  Lim  A ) )
9581, 94mpbid 210 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  Lim  A )
96 cflm 8630 . . . . . . . 8  |-  ( ( A  e.  On  /\  Lim  A )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
9732, 95, 96syl2anc 661 . . . . . . 7  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
98 cardon 8325 . . . . . . . . . . . 12  |-  ( card `  y )  e.  On
99 eleq1 2539 . . . . . . . . . . . 12  |-  ( x  =  ( card `  y
)  ->  ( x  e.  On  <->  ( card `  y
)  e.  On ) )
10098, 99mpbiri 233 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  x  e.  On )
101100adantr 465 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  x  e.  On )
102101exlimiv 1698 . . . . . . . . 9  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  ->  x  e.  On )
103102abssi 3575 . . . . . . . 8  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On
104 fvex 5876 . . . . . . . . . 10  |-  ( cf `  A )  e.  _V
10597, 104syl6eqelr 2564 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
106 intex 4603 . . . . . . . . 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 212 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  =/=  (/) )
108 onint 6614 . . . . . . . 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 663 . . . . . . 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 2555 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  e. 
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
111 eqeq1 2471 . . . . . . . . 9  |-  ( x  =  ( cf `  A
)  ->  ( x  =  ( card `  y
)  <->  ( cf `  A
)  =  ( card `  y ) ) )
112111anbi1d 704 . . . . . . . 8  |-  ( x  =  ( cf `  A
)  ->  ( (
x  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  <->  ( ( cf `  A )  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) ) )
113112exbidv 1690 . . . . . . 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 3250 . . . . . 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 196 . . . . 5  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  E. y
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) ) )
116 simp2rr 1066 . . . . . . . 8  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  =  U. y )
117 simp1l 1020 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  U  e.  Univ )
118 simp2rl 1065 . . . . . . . . . . 11  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  A )
119118, 43syl6ss 3516 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  U )
12043sseli 3500 . . . . . . . . . . 11  |-  ( ( cf `  A )  e.  A  ->  ( cf `  A )  e.  U )
1211203ad2ant3 1019 . . . . . . . . . 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 1022 . . . . . . . . . . 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 3116 . . . . . . . . . . . 12  |-  y  e. 
_V
124123cardid 8922 . . . . . . . . . . 11  |-  ( card `  y )  ~~  y
125122, 124syl6eqbr 4484 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  ~~  y )
126 gruen 9190 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  y  C_  U  /\  ( ( cf `  A )  e.  U  /\  ( cf `  A )  ~~  y ) )  -> 
y  e.  U )
127117, 119, 121, 125, 126syl112anc 1232 . . . . . . . . 9  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  e.  U )
128 gruuni 9178 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  y  e.  U )  ->  U. y  e.  U )
129117, 127, 128syl2anc 661 . . . . . . . 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 2555 . . . . . . 7  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  A  e.  U )
1311303exp 1195 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  (
( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  ->  (
( cf `  A
)  e.  A  ->  A  e.  U )
) )
132131exlimdv 1700 . . . . 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 177 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  -.  ( cf `  A )  e.  A )
135 cfon 8635 . . . . 5  |-  ( cf `  A )  e.  On
136 cfle 8634 . . . . . 6  |-  ( cf `  A )  C_  A
137 onsseleq 4919 . . . . . 6  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  C_  A  <->  ( ( cf `  A )  e.  A  \/  ( cf `  A )  =  A ) ) )
138136, 137mpbii 211 . . . . 5  |-  ( ( ( cf `  A
)  e.  On  /\  A  e.  On )  ->  ( ( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
139135, 138mpan 670 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  e.  A  \/  ( cf `  A )  =  A ) )
140139ord 377 . . 3  |-  ( A  e.  On  ->  ( -.  ( cf `  A
)  e.  A  -> 
( cf `  A
)  =  A ) )
14132, 134, 140sylc 60 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  =  A )
14276adantr 465 . 2  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A. x  e.  A  ~P x  ~<  A )
143 elina 9065 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
14416, 141, 142, 143syl3anbrc 1180 1  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   class class class wbr 4447   Tr wtr 4540    _E cep 4789    We wwe 4837   Ord word 4877   Oncon0 4878   Lim wlim 4879   ` cfv 5588   omcom 6684    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   cardccrd 8316   cfccf 8318   Inacccina 9061   Univcgru 9168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-ac2 8843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-1o 7130  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-card 8320  df-cf 8322  df-ac 8497  df-ina 9063  df-gru 9169
This theorem is referenced by:  grur1a  9197  grur1  9198  grutsk  9200
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