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Theorem gruina 8626
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 3580 . . . 4  |-  ( U  =/=  (/)  <->  E. x  x  e.  U )
2 0ss 3599 . . . . . . . . . . 11  |-  (/)  C_  x
3 gruss 8604 . . . . . . . . . . 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 4575 . . . . . . . . . 10  |-  (/)  e.  On
64, 5jctir 525 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ( (/) 
e.  U  /\  (/)  e.  On ) )
7 elin 3473 . . . . . . . . 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 2478 . . . . . . 7  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  (/)  e.  A
)
11 ne0i 3577 . . . . . . 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 1641 . . . 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 8601 . . . . . . . 8  |-  ( U  e.  Univ  ->  Tr  U
)
18 tron 4545 . . . . . . . 8  |-  Tr  On
19 trin 4253 . . . . . . . 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 3505 . . . . . . . . 9  |-  ( U  i^i  On )  C_  On
22 epweon 4704 . . . . . . . . 9  |-  _E  We  On
23 wess 4510 . . . . . . . . 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 4525 . . . . . . 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 4287 . . . . . 6  |-  ( U  e.  Univ  ->  ( U  i^i  On )  e. 
_V )
29 elon2 4533 . . . . . 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 2471 . . . 4  |-  ( U  e.  Univ  ->  A  e.  On )
3231adantr 452 . . 3  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  On )
33 eloni 4532 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
34 ordirr 4540 . . . . . . 7  |-  ( Ord 
A  ->  -.  A  e.  A )
3533, 34syl 16 . . . . . 6  |-  ( A  e.  On  ->  -.  A  e.  A )
36 elin 3473 . . . . . . . . 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 2478 . . . . . . 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 3504 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  On )  C_  U
439, 42eqsstri 3321 . . . . . . . . . . . . . . 15  |-  A  C_  U
4443sseli 3287 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  x  e.  U )
45 vex 2902 . . . . . . . . . . . . . . . . 17  |-  x  e. 
_V
4645pwex 4323 . . . . . . . . . . . . . . . 16  |-  ~P x  e.  _V
4746canth2 7196 . . . . . . . . . . . . . . 15  |-  ~P x  ~<  ~P ~P x
4846pwex 4323 . . . . . . . . . . . . . . . . . 18  |-  ~P ~P x  e.  _V
4948cardid 8355 . . . . . . . . . . . . . . . . 17  |-  ( card `  ~P ~P x ) 
~~  ~P ~P x
5049ensymi 7093 . . . . . . . . . . . . . . . 16  |-  ~P ~P x  ~~  ( card `  ~P ~P x )
5131adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  A  e.  On )
52 grupw 8603 . . . . . . . . . . . . . . . . . . 19  |-  ( ( U  e.  Univ  /\  x  e.  U )  ->  ~P x  e.  U )
53 grupw 8603 . . . . . . . . . . . . . . . . . . 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 7070 . . . . . . . . . . . . . . . . . . . . . 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 7764 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( card `  ~P ~P x )  e.  On
59 grudomon 8625 . . . . . . . . . . . . . . . . . . . . . 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 3473 . . . . . . . . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . . . . . . . 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 4564 . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . 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 7101 . . . . . . . . . . . . . . . 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 7176 . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  ~P x  ~<  A )
77 inawinalem 8497 . . . . . . . . . . . 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 8501 . . . . . . . . . 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 7196 . . . . . . . . . . . . . 14  |-  x  ~<  ~P x
83 sdomtr 7181 . . . . . . . . . . . . . 14  |-  ( ( x  ~<  ~P x  /\  ~P x  ~<  A )  ->  x  ~<  A )
8482, 75, 83sylancr 645 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  x  e.  A )  ->  x  ~<  A )
8584ralrimiva 2732 . . . . . . . . . . . 12  |-  ( U  e.  Univ  ->  A. x  e.  A  x  ~<  A )
86 iscard 7795 . . . . . . . . . . . 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 7792 . . . . . . . . . . . 12  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
89 sseq2 3313 . . . . . . . . . . . . 13  |-  ( (
card `  A )  =  A  ->  ( om  C_  ( card `  A
)  <->  om  C_  A )
)
90 limeq 4534 . . . . . . . . . . . . 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 8063 . . . . . . . 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 7764 . . . . . . . . . . . 12  |-  ( card `  y )  e.  On
99 eleq1 2447 . . . . . . . . . . . 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 1641 . . . . . . . . 9  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) )  ->  x  e.  On )
103102abssi 3361 . . . . . . . 8  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  C_  On
104 fvex 5682 . . . . . . . . . 10  |-  ( cf `  A )  e.  _V
10597, 104syl6eqelr 2476 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) }  e.  _V )
106 intex 4297 . . . . . . . . 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 4715 . . . . . . . 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 2461 . . . . . 6  |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  ( cf `  A )  e. 
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A  = 
U. y ) ) } )
111 eqeq1 2393 . . . . . . . . 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 1633 . . . . . . 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 3025 . . . . . 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 3303 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  y  C_  U )
12043sseli 3287 . . . . . . . . . . 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 2902 . . . . . . . . . . . 12  |-  y  e. 
_V
124123cardid 8355 . . . . . . . . . . 11  |-  ( card `  y )  ~~  y
125122, 124syl6eqbr 4190 . . . . . . . . . 10  |-  ( ( ( U  e.  Univ  /\  U  =/=  (/) )  /\  ( ( cf `  A
)  =  ( card `  y )  /\  (
y  C_  A  /\  A  =  U. y
) )  /\  ( cf `  A )  e.  A )  ->  ( cf `  A )  ~~  y )
126 gruen 8620 . . . . . . . . . 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 8608 . . . . . . . . 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 2461 . . . . . . 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 1643 . . . . 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 8068 . . . . 5  |-  ( cf `  A )  e.  On
136 cfle 8067 . . . . . 6  |-  ( cf `  A )  C_  A
137 onsseleq 4563 . . . . . 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 8495 . 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 1547    = wceq 1649    e. wcel 1717   {cab 2373    =/= wne 2550   A.wral 2649   E.wrex 2650   _Vcvv 2899    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   class class class wbr 4153   Tr wtr 4243    _E cep 4433    We wwe 4481   Ord word 4521   Oncon0 4522   Lim wlim 4523   omcom 4785   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755   cfccf 7757   Inacccina 8491   Univcgru 8598
This theorem is referenced by:  grur1a  8627  grur1  8628  grutsk  8630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-ac2 8276
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-1o 6660  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-card 7759  df-cf 7761  df-ac 7930  df-ina 8493  df-gru 8599
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