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Theorem heiborlem4 26641
Description: Lemma for heibor 26648. Using the function  T constructed in heiborlem3 26640, construct an infinite path in  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
heibor.9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
heibor.10  |-  ( ph  ->  C G 0 )
heibor.11  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
Assertion
Ref Expression
heiborlem4  |-  ( (
ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
Distinct variable groups:    x, n, y, A    u, n, F, x, y    x, G    ph, x    m, n, u, v, x, y, z, D    T, m, n, x, y, z    B, n, u, v, y    m, J, n, u, v, x, y, z    U, n, u, v, x, y, z    S, m, n, u, v, x, y, z   
m, X, n, u, v, x, y, z    C, m, n, u, v, y    n, K, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    A( z, v, u, m)    B( z, m)    C( x, z)    T( v, u)    U( m)    F( z, v, m)    G( y, z, v, u, m, n)    K( v, u, m)

Proof of Theorem heiborlem4
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . 5  |-  ( x  =  0  ->  ( S `  x )  =  ( S ` 
0 ) )
2 id 19 . . . . 5  |-  ( x  =  0  ->  x  =  0 )
31, 2breq12d 4052 . . . 4  |-  ( x  =  0  ->  (
( S `  x
) G x  <->  ( S `  0 ) G 0 ) )
43imbi2d 307 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  0
) G 0 ) ) )
5 fveq2 5541 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
6 id 19 . . . . 5  |-  ( x  =  k  ->  x  =  k )
75, 6breq12d 4052 . . . 4  |-  ( x  =  k  ->  (
( S `  x
) G x  <->  ( S `  k ) G k ) )
87imbi2d 307 . . 3  |-  ( x  =  k  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  k
) G k ) ) )
9 fveq2 5541 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( S `  x )  =  ( S `  ( k  +  1 ) ) )
10 id 19 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  x  =  ( k  +  1 ) )
119, 10breq12d 4052 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( S `  x
) G x  <->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
1211imbi2d 307 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
13 fveq2 5541 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
14 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1513, 14breq12d 4052 . . . 4  |-  ( x  =  A  ->  (
( S `  x
) G x  <->  ( S `  A ) G A ) )
1615imbi2d 307 . . 3  |-  ( x  =  A  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  A
) G A ) ) )
17 heibor.11 . . . . . . 7  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
1817fveq1i 5542 . . . . . 6  |-  ( S `
 0 )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  0
)
19 0z 10051 . . . . . . 7  |-  0  e.  ZZ
20 seq1 11075 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) ` 
0 )  =  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  0
) )
2119, 20ax-mp 8 . . . . . 6  |-  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) ` 
0 )  =  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  0
)
2218, 21eqtri 2316 . . . . 5  |-  ( S `
 0 )  =  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )
23 0nn0 9996 . . . . . 6  |-  0  e.  NN0
24 heibor.10 . . . . . . 7  |-  ( ph  ->  C G 0 )
25 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
2625relopabi 4827 . . . . . . . 8  |-  Rel  G
2726brrelexi 4745 . . . . . . 7  |-  ( C G 0  ->  C  e.  _V )
2824, 27syl 15 . . . . . 6  |-  ( ph  ->  C  e.  _V )
29 iftrue 3584 . . . . . . 7  |-  ( m  =  0  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  C )
30 eqid 2296 . . . . . . 7  |-  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )  =  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )
3129, 30fvmptg 5616 . . . . . 6  |-  ( ( 0  e.  NN0  /\  C  e.  _V )  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3223, 28, 31sylancr 644 . . . . 5  |-  ( ph  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3322, 32syl5eq 2340 . . . 4  |-  ( ph  ->  ( S `  0
)  =  C )
3433, 24eqbrtrd 4059 . . 3  |-  ( ph  ->  ( S `  0
) G 0 )
35 df-br 4040 . . . . . 6  |-  ( ( S `  k ) G k  <->  <. ( S `
 k ) ,  k >.  e.  G
)
36 heibor.9 . . . . . . 7  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
37 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( T `  <. ( S `  k ) ,  k >. )
)
38 df-ov 5877 . . . . . . . . . . 11  |-  ( ( S `  k ) T k )  =  ( T `  <. ( S `  k ) ,  k >. )
3937, 38syl6eqr 2346 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( ( S `  k
) T k ) )
40 fvex 5555 . . . . . . . . . . . 12  |-  ( S `
 k )  e. 
_V
41 vex 2804 . . . . . . . . . . . 12  |-  k  e. 
_V
4240, 41op2ndd 6147 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( 2nd `  x
)  =  k )
4342oveq1d 5889 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( 2nd `  x )  +  1 )  =  ( k  +  1 ) )
4439, 43breq12d 4052 . . . . . . . . 9  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) G ( ( 2nd `  x
)  +  1 )  <-> 
( ( S `  k ) T k ) G ( k  +  1 ) ) )
45 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( B `  <. ( S `  k ) ,  k >. )
)
46 df-ov 5877 . . . . . . . . . . . 12  |-  ( ( S `  k ) B k )  =  ( B `  <. ( S `  k ) ,  k >. )
4745, 46syl6eqr 2346 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( ( S `  k
) B k ) )
4839, 43oveq12d 5892 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) )  =  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) )
4947, 48ineq12d 3384 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  =  ( ( ( S `  k ) B k )  i^i  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) ) )
5049eleq1d 2362 . . . . . . . . 9  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( ( B `  x )  i^i  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K  <->  ( ( ( S `  k ) B k )  i^i  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) )  e.  K ) )
5144, 50anbi12d 691 . . . . . . . 8  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  <->  ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k
) B k )  i^i  ( ( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
5251rspccv 2894 . . . . . . 7  |-  ( A. x  e.  G  (
( T `  x
) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
( T `  x
) B ( ( 2nd `  x )  +  1 ) ) )  e.  K )  ->  ( <. ( S `  k ) ,  k >.  e.  G  ->  ( ( ( S `
 k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k ) B k )  i^i  ( ( ( S `
 k ) T k ) B ( k  +  1 ) ) )  e.  K
) ) )
5336, 52syl 15 . . . . . 6  |-  ( ph  ->  ( <. ( S `  k ) ,  k
>.  e.  G  ->  (
( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `
 k ) B k )  i^i  (
( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
5435, 53syl5bi 208 . . . . 5  |-  ( ph  ->  ( ( S `  k ) G k  ->  ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k
) B k )  i^i  ( ( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
55 seqp1 11077 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  0
)  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  ( k  +  1 ) )  =  ( (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) ) )
56 nn0uz 10278 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
5755, 56eleq2s 2388 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  ( k  +  1 ) )  =  ( (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) ) )
5817fveq1i 5542 . . . . . . . . . 10  |-  ( S `
 ( k  +  1 ) )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  (
k  +  1 ) )
5917fveq1i 5542 . . . . . . . . . . 11  |-  ( S `
 k )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
)
6059oveq1i 5884 . . . . . . . . . 10  |-  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )  =  ( (  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) ) `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )
6157, 58, 603eqtr4g 2353 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  (
k  +  1 ) ) ) )
62 peano2nn0 10020 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( k  +  1 )  e. 
NN0 )
63 nn0p1nn 10019 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
64 nnne0 9794 . . . . . . . . . . . . . . 15  |-  ( ( k  +  1 )  e.  NN  ->  (
k  +  1 )  =/=  0 )
6564neneqd 2475 . . . . . . . . . . . . . 14  |-  ( ( k  +  1 )  e.  NN  ->  -.  ( k  +  1 )  =  0 )
66 iffalse 3585 . . . . . . . . . . . . . 14  |-  ( -.  ( k  +  1 )  =  0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  -  1 ) )
6763, 65, 663syl 18 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  - 
1 ) )
68 ovex 5899 . . . . . . . . . . . . 13  |-  ( ( k  +  1 )  -  1 )  e. 
_V
6967, 68syl6eqel 2384 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  e.  _V )
70 eqeq1 2302 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  =  0  <->  (
k  +  1 )  =  0 ) )
71 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  -  1 )  =  ( ( k  +  1 )  - 
1 ) )
7270, 71ifbieq2d 3598 . . . . . . . . . . . . 13  |-  ( m  =  ( k  +  1 )  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  - 
1 ) ) )
7372, 30fvmptg 5616 . . . . . . . . . . . 12  |-  ( ( ( k  +  1 )  e.  NN0  /\  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  e.  _V )  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 ( k  +  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C , 
( ( k  +  1 )  -  1 ) ) )
7462, 69, 73syl2anc 642 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  - 
1 ) ) )
75 nn0cn 9991 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
76 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
77 pncan 9073 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  +  1 )  -  1 )  =  k )
7875, 76, 77sylancl 643 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( k  +  1 )  -  1 )  =  k )
7974, 67, 783eqtrd 2332 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) )  =  k )
8079oveq2d 5890 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )  =  ( ( S `  k ) T k ) )
8161, 80eqtrd 2328 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T k ) )
8281breq1d 4049 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( S `  ( k  +  1 ) ) G ( k  +  1 )  <->  ( ( S `  k ) T k ) G ( k  +  1 ) ) )
8382biimprd 214 . . . . . 6  |-  ( k  e.  NN0  ->  ( ( ( S `  k
) T k ) G ( k  +  1 )  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
8483adantrd 454 . . . . 5  |-  ( k  e.  NN0  ->  ( ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `
 k ) B k )  i^i  (
( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K )  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
8554, 84syl9r 67 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( S `  k ) G k  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) ) )
8685a2d 23 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( S `  k ) G k )  ->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
874, 8, 12, 16, 34, 86nn0ind 10124 . 2  |-  ( A  e.  NN0  ->  ( ph  ->  ( S `  A
) G A ) )
8887impcom 419 1  |-  ( (
ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   Fincfn 6879   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   ^cexp 11120   ballcbl 16387   MetOpencmopn 16388   CMetcms 18696
This theorem is referenced by:  heiborlem5  26642  heiborlem6  26643  heiborlem8  26645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063
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