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Theorem lecldbas 18723
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 2441 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  =  ran  ( y  e.  RR*  |->  ( y (,] +oo ) )
2 eqid 2441 . . . 4  |-  ran  (
y  e.  RR*  |->  ( -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  ( -oo [,) y ) )
31, 2leordtval2 18716 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) )
4 fvex 5698 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5698 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 11384 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5556 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 5 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6238 . . . . . . . . . 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 3465 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 18718 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 18711 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 18535 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2529 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2835 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2835 . . . . . . . . 9  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x
)  e.  (ordTop `  <_  ) )
2011, 19sylbi 195 . . . . . . . 8  |-  ( x  e.  ran  [,]  ->  (
RR*  \  x )  e.  (ordTop `  <_  ) )
216, 20fmpti 5863 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5562 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 5 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4434 . . . . 5  |-  ran  F  e.  _V
25 eqid 2441 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,] +oo ) )  =  ( y  e. 
RR*  |->  ( y (,] +oo ) )
26 mnfxr 11090 . . . . . . . . . . 11  |- -oo  e.  RR*
27 fnovrn 6237 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\  y  e. 
RR* )  ->  ( -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1299 . . . . . . . . . 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 11088 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> +oo  e.  RR* )
33 mnfle 11109 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> -oo  <_  y )
34 pnfge 11106 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_ +oo )
35 df-icc 11303 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 11301 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9439 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 11128 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
39 xrlelttr 11126 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <  z ) )
40 xrltle 11122 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  e.  RR*  /\  z  e.  RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
41403adant2 1002 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
4239, 41syld 44 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <_  z ) )
4335, 36, 37, 35, 38, 42ixxun 11312 . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  ( -oo [,] +oo )
)
45 iccmax 11367 . . . . . . . . . . . . 13  |-  ( -oo [,] +oo )  =  RR*
4644, 45syl6eq 2489 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  RR* )
47 iccssxr 11374 . . . . . . . . . . . . 13  |-  ( -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 11311 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
4926, 31, 48mp3an13 1300 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
50 uneqdifeq 3764 . . . . . . . . . . . . 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 658 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( -oo [,] y
)  u.  ( y (,] +oo ) )  =  RR*  <->  ( RR*  \  ( -oo [,] y ) )  =  ( y (,] +oo ) ) )
5246, 51mpbid 210 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( -oo [,] y
) )  =  ( y (,] +oo )
)
5352eqcomd 2446 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y ) ) )
54 difeq2 3465 . . . . . . . . . . . 12  |-  ( x  =  ( -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  ( -oo [,] y ) ) )
5554eqeq2d 2452 . . . . . . . . . . 11  |-  ( x  =  ( -oo [,] y )  ->  (
( y (,] +oo )  =  ( RR*  \  x )  <->  ( y (,] +oo )  =  (
RR*  \  ( -oo [,] y ) ) ) )
5655rspcev 3070 . . . . . . . . . 10  |-  ( ( ( -oo [,] y
)  e.  ran  [,]  /\  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y
) ) )  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x ) )
5728, 53, 56syl2anc 656 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x
) )
58 xrex 10984 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4437 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 5 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5086 . . . . . . . . 9  |-  ( ( y (,] +oo )  e.  ran  F  <->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x
) )
6257, 61sylibr 212 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( y (,] +oo )  e. 
ran  F )
6325, 62fmpti 5863 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,] +oo ) ) : RR* --> ran  F
64 frn 5562 . . . . . . 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 2441 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( -oo [,) y ) )  =  ( y  e.  RR*  |->  ( -oo [,) y ) )
67 fnovrn 6237 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( y [,] +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1300 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,] +oo )  e. 
ran  [,] )
69 df-ico 11302 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9438 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 11127 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  < +oo ) )
72 xrltle 11122 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\ +oo  e.  RR* )  ->  (
z  < +oo  ->  z  <_ +oo ) )
73723adant2 1002 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( z  < +oo  ->  z  <_ +oo ) )
7471, 73syld 44 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
75 xrletr 11128 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <_  z )  -> -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 11312 . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( -oo [,] +oo )
)
78 uncom 3497 . . . . . . . . . . . . 13  |-  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( ( y [,] +oo )  u.  ( -oo [,) y ) )
7977, 78, 453eqtr3g 2496 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  u.  ( -oo [,) y
) )  =  RR* )
80 iccssxr 11374 . . . . . . . . . . . . 13  |-  ( y [,] +oo )  C_  RR*
81 incom 3540 . . . . . . . . . . . . . 14  |-  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  ( ( -oo [,) y
)  i^i  ( y [,] +oo ) )
8269, 35, 70ixxdisj 11311 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8326, 31, 82mp3an13 1300 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8481, 83syl5eq 2485 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  (/) )
85 uneqdifeq 3764 . . . . . . . . . . . . 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 658 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( y [,] +oo )  u.  ( -oo [,) y ) )  = 
RR* 
<->  ( RR*  \  (
y [,] +oo )
)  =  ( -oo [,) y ) ) )
8779, 86mpbid 210 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( y [,] +oo ) )  =  ( -oo [,) y ) )
8887eqcomd 2446 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) )
89 difeq2 3465 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] +oo )  ->  ( RR*  \  x )  =  (
RR*  \  ( y [,] +oo ) ) )
9089eqeq2d 2452 . . . . . . . . . . 11  |-  ( x  =  ( y [,] +oo )  ->  ( ( -oo [,) y )  =  ( RR*  \  x
)  <->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) ) )
9190rspcev 3070 . . . . . . . . . 10  |-  ( ( ( y [,] +oo )  e.  ran  [,]  /\  ( -oo [,) y )  =  ( RR*  \  (
y [,] +oo )
) )  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
9268, 88, 91syl2anc 656 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
936, 60elrnmpti 5086 . . . . . . . . 9  |-  ( ( -oo [,) y )  e.  ran  F  <->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
9492, 93sylibr 212 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( -oo [,) y )  e.  ran  F )
9566, 94fmpti 5863 . . . . . . 7  |-  ( y  e.  RR*  |->  ( -oo [,) y ) ) :
RR* --> ran  F
96 frn 5562 . . . . . . 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 3528 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) )  C_  ran  F
99 fiss 7670 . . . . 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 667 . . . 4  |-  ( fi
`  ( ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) )  C_  ( fi ` 
ran  F )
101 tgss 18473 . . . 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 667 . . 3  |-  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) )
1033, 102eqsstri 3383 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 18710 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 18496 . . 3  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ran  F  C_  (ordTop `  <_  ) )  ->  ( topGen `  ( fi ` 
ran  F ) ) 
C_  (ordTop `  <_  ) )
106104, 23, 105mp2an 667 . 2  |-  ( topGen `  ( fi `  ran  F ) )  C_  (ordTop ` 
<_  )
107103, 106eqssi 3369 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   _Vcvv 2970    \ cdif 3322    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   ficfi 7656   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413    < clt 9414    <_ cle 9415   (,]cioc 11297   [,)cico 11298   [,]cicc 11299   topGenctg 14372  ordTopcordt 14433   Topctop 18398   Clsdccld 18520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-ioc 11301  df-ico 11302  df-icc 11303  df-topgen 14378  df-ordt 14435  df-ps 15366  df-tsr 15367  df-top 18403  df-bases 18405  df-topon 18406  df-cld 18523
This theorem is referenced by: (None)
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