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Theorem lecldbas 20247
Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
lecldbas.1  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
Assertion
Ref Expression
lecldbas  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )

Proof of Theorem lecldbas
Dummy variables  a 
b  c  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2453 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  =  ran  ( y  e.  RR*  |->  ( y (,] +oo ) )
2 eqid 2453 . . . 4  |-  ran  (
y  e.  RR*  |->  ( -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  ( -oo [,) y ) )
31, 2leordtval2 20240 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) )
4 fvex 5880 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5880 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 11740 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5733 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 5 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6450 . . . . . . . . . 10  |-  ( [,] 
Fn  ( RR*  X.  RR* )  ->  ( x  e. 
ran  [,]  <->  E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b ) ) )
119, 10ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ran  [,]  <->  E. a  e.  RR*  E. b  e. 
RR*  x  =  ( a [,] b ) )
12 difeq2 3547 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 20242 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 20235 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 20058 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2539 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2878 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2878 . . . . . . . . 9  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x
)  e.  (ordTop `  <_  ) )
2011, 19sylbi 199 . . . . . . . 8  |-  ( x  e.  ran  [,]  ->  (
RR*  \  x )  e.  (ordTop `  <_  ) )
216, 20fmpti 6050 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5740 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 5 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4551 . . . . 5  |-  ran  F  e.  _V
25 eqid 2453 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,] +oo ) )  =  ( y  e. 
RR*  |->  ( y (,] +oo ) )
26 mnfxr 11421 . . . . . . . . . . 11  |- -oo  e.  RR*
27 fnovrn 6449 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\  y  e. 
RR* )  ->  ( -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1356 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( -oo [,] y )  e.  ran  [,] )
2926a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> -oo  e.  RR* )
30 id 22 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  e. 
RR* )
31 pnfxr 11419 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> +oo  e.  RR* )
33 mnfle 11442 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> -oo  <_  y )
34 pnfge 11439 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_ +oo )
35 df-icc 11649 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 11647 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9706 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 11462 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
39 xrlelttr 11460 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <  z ) )
40 xrltle 11455 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  e.  RR*  /\  z  e.  RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
41403adant2 1028 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
4239, 41syld 45 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <_  z ) )
4335, 36, 37, 35, 38, 42ixxun 11658 . . . . . . . . . . . . . 14  |-  ( ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  y  /\  y  <_ +oo ) )  -> 
( ( -oo [,] y )  u.  (
y (,] +oo )
)  =  ( -oo [,] +oo ) )
4429, 30, 32, 33, 34, 43syl32anc 1277 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  ( -oo [,] +oo )
)
45 iccmax 11717 . . . . . . . . . . . . 13  |-  ( -oo [,] +oo )  =  RR*
4644, 45syl6eq 2503 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  RR* )
47 iccssxr 11724 . . . . . . . . . . . . 13  |-  ( -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 11657 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
4926, 31, 48mp3an13 1357 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
50 uneqdifeq 3858 . . . . . . . . . . . . 13  |-  ( ( ( -oo [,] y
)  C_  RR*  /\  (
( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )  ->  ( ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  RR*  <->  ( RR*  \  ( -oo [,] y ) )  =  ( y (,] +oo ) ) )
5147, 49, 50sylancr 670 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( -oo [,] y
)  u.  ( y (,] +oo ) )  =  RR*  <->  ( RR*  \  ( -oo [,] y ) )  =  ( y (,] +oo ) ) )
5246, 51mpbid 214 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( -oo [,] y
) )  =  ( y (,] +oo )
)
5352eqcomd 2459 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y ) ) )
54 difeq2 3547 . . . . . . . . . . . 12  |-  ( x  =  ( -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  ( -oo [,] y ) ) )
5554eqeq2d 2463 . . . . . . . . . . 11  |-  ( x  =  ( -oo [,] y )  ->  (
( y (,] +oo )  =  ( RR*  \  x )  <->  ( y (,] +oo )  =  (
RR*  \  ( -oo [,] y ) ) ) )
5655rspcev 3152 . . . . . . . . . 10  |-  ( ( ( -oo [,] y
)  e.  ran  [,]  /\  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y
) ) )  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x ) )
5728, 53, 56syl2anc 667 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x
) )
58 xrex 11306 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4554 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 5 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5088 . . . . . . . . 9  |-  ( ( y (,] +oo )  e.  ran  F  <->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x
) )
6257, 61sylibr 216 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( y (,] +oo )  e. 
ran  F )
6325, 62fmpti 6050 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,] +oo ) ) : RR* --> ran  F
64 frn 5740 . . . . . . 7  |-  ( ( y  e.  RR*  |->  ( y (,] +oo ) ) : RR* --> ran  F  ->  ran  ( y  e. 
RR*  |->  ( y (,] +oo ) )  C_  ran  F )
6563, 64ax-mp 5 . . . . . 6  |-  ran  (
y  e.  RR*  |->  ( y (,] +oo ) ) 
C_  ran  F
66 eqid 2453 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( -oo [,) y ) )  =  ( y  e.  RR*  |->  ( -oo [,) y ) )
67 fnovrn 6449 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( y [,] +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1357 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,] +oo )  e. 
ran  [,] )
69 df-ico 11648 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9704 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 11461 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  < +oo ) )
72 xrltle 11455 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\ +oo  e.  RR* )  ->  (
z  < +oo  ->  z  <_ +oo ) )
73723adant2 1028 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( z  < +oo  ->  z  <_ +oo ) )
7471, 73syld 45 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
75 xrletr 11462 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <_  z )  -> -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 11658 . . . . . . . . . . . . . 14  |-  ( ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  y  /\  y  <_ +oo ) )  -> 
( ( -oo [,) y )  u.  (
y [,] +oo )
)  =  ( -oo [,] +oo ) )
7729, 30, 32, 33, 34, 76syl32anc 1277 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( -oo [,] +oo )
)
78 uncom 3580 . . . . . . . . . . . . 13  |-  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( ( y [,] +oo )  u.  ( -oo [,) y ) )
7977, 78, 453eqtr3g 2510 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  u.  ( -oo [,) y
) )  =  RR* )
80 iccssxr 11724 . . . . . . . . . . . . 13  |-  ( y [,] +oo )  C_  RR*
81 incom 3627 . . . . . . . . . . . . . 14  |-  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  ( ( -oo [,) y
)  i^i  ( y [,] +oo ) )
8269, 35, 70ixxdisj 11657 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8326, 31, 82mp3an13 1357 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8481, 83syl5eq 2499 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  (/) )
85 uneqdifeq 3858 . . . . . . . . . . . . 13  |-  ( ( ( y [,] +oo )  C_  RR*  /\  (
( y [,] +oo )  i^i  ( -oo [,) y ) )  =  (/) )  ->  ( ( ( y [,] +oo )  u.  ( -oo [,) y ) )  = 
RR* 
<->  ( RR*  \  (
y [,] +oo )
)  =  ( -oo [,) y ) ) )
8680, 84, 85sylancr 670 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( y [,] +oo )  u.  ( -oo [,) y ) )  = 
RR* 
<->  ( RR*  \  (
y [,] +oo )
)  =  ( -oo [,) y ) ) )
8779, 86mpbid 214 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( y [,] +oo ) )  =  ( -oo [,) y ) )
8887eqcomd 2459 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) )
89 difeq2 3547 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] +oo )  ->  ( RR*  \  x )  =  (
RR*  \  ( y [,] +oo ) ) )
9089eqeq2d 2463 . . . . . . . . . . 11  |-  ( x  =  ( y [,] +oo )  ->  ( ( -oo [,) y )  =  ( RR*  \  x
)  <->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) ) )
9190rspcev 3152 . . . . . . . . . 10  |-  ( ( ( y [,] +oo )  e.  ran  [,]  /\  ( -oo [,) y )  =  ( RR*  \  (
y [,] +oo )
) )  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
9268, 88, 91syl2anc 667 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
936, 60elrnmpti 5088 . . . . . . . . 9  |-  ( ( -oo [,) y )  e.  ran  F  <->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
9492, 93sylibr 216 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( -oo [,) y )  e.  ran  F )
9566, 94fmpti 6050 . . . . . . 7  |-  ( y  e.  RR*  |->  ( -oo [,) y ) ) :
RR* --> ran  F
96 frn 5740 . . . . . . 7  |-  ( ( y  e.  RR*  |->  ( -oo [,) y ) ) :
RR* --> ran  F  ->  ran  ( y  e.  RR*  |->  ( -oo [,) y ) )  C_  ran  F )
9795, 96ax-mp 5 . . . . . 6  |-  ran  (
y  e.  RR*  |->  ( -oo [,) y ) )  C_  ran  F
9865, 97unssi 3611 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) )  C_  ran  F
99 fiss 7943 . . . . 5  |-  ( ( ran  F  e.  _V  /\  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) 
C_  ran  F )  ->  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) )  C_  ( fi `  ran  F
) )
10024, 98, 99mp2an 679 . . . 4  |-  ( fi
`  ( ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) )  C_  ( fi ` 
ran  F )
101 tgss 19996 . . . 4  |-  ( ( ( fi `  ran  F )  e.  _V  /\  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) )  C_  ( fi `  ran  F
) )  ->  ( topGen `
 ( fi `  ( ran  ( y  e. 
RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) ) )
1024, 100, 101mp2an 679 . . 3  |-  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) )
1033, 102eqsstri 3464 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 20234 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 20019 . . 3  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ran  F  C_  (ordTop `  <_  ) )  ->  ( topGen `  ( fi ` 
ran  F ) ) 
C_  (ordTop `  <_  ) )
106104, 23, 105mp2an 679 . 2  |-  ( topGen `  ( fi `  ran  F ) )  C_  (ordTop ` 
<_  )
107103, 106eqssi 3450 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   E.wrex 2740   _Vcvv 3047    \ cdif 3403    u. cun 3404    i^i cin 3405    C_ wss 3406   (/)c0 3733   ~Pcpw 3953   class class class wbr 4405    |-> cmpt 4464    X. cxp 4835   ran crn 4838    Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   ficfi 7929   +oocpnf 9677   -oocmnf 9678   RR*cxr 9679    < clt 9680    <_ cle 9681   (,]cioc 11643   [,)cico 11644   [,]cicc 11645   topGenctg 15348  ordTopcordt 15409   Topctop 19929   Clsdccld 20043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-ioc 11647  df-ico 11648  df-icc 11649  df-topgen 15354  df-ordt 15411  df-ps 16458  df-tsr 16459  df-top 19933  df-bases 19934  df-topon 19935  df-cld 20046
This theorem is referenced by: (None)
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