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Theorem lecldbas 18835
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 2443 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  =  ran  ( y  e.  RR*  |->  ( y (,] +oo ) )
2 eqid 2443 . . . 4  |-  ran  (
y  e.  RR*  |->  ( -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  ( -oo [,) y ) )
31, 2leordtval2 18828 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) )
4 fvex 5713 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5713 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 11400 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5571 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 5 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6251 . . . . . . . . . 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 3480 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 18830 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 18823 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 18647 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2531 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2849 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2849 . . . . . . . . 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 5878 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5577 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 5 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4449 . . . . 5  |-  ran  F  e.  _V
25 eqid 2443 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,] +oo ) )  =  ( y  e. 
RR*  |->  ( y (,] +oo ) )
26 mnfxr 11106 . . . . . . . . . . 11  |- -oo  e.  RR*
27 fnovrn 6250 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\  y  e. 
RR* )  ->  ( -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1304 . . . . . . . . . 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 11104 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> +oo  e.  RR* )
33 mnfle 11125 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> -oo  <_  y )
34 pnfge 11122 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_ +oo )
35 df-icc 11319 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 11317 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9455 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 11144 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
39 xrlelttr 11142 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <  z ) )
40 xrltle 11138 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  e.  RR*  /\  z  e.  RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
41403adant2 1007 . . . . . . . . . . . . . . . 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 11328 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  ( -oo [,] +oo )
)
45 iccmax 11383 . . . . . . . . . . . . 13  |-  ( -oo [,] +oo )  =  RR*
4644, 45syl6eq 2491 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  RR* )
47 iccssxr 11390 . . . . . . . . . . . . 13  |-  ( -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 11327 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
4926, 31, 48mp3an13 1305 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
50 uneqdifeq 3779 . . . . . . . . . . . . 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 663 . . . . . . . . . . . 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 2448 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y ) ) )
54 difeq2 3480 . . . . . . . . . . . 12  |-  ( x  =  ( -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  ( -oo [,] y ) ) )
5554eqeq2d 2454 . . . . . . . . . . 11  |-  ( x  =  ( -oo [,] y )  ->  (
( y (,] +oo )  =  ( RR*  \  x )  <->  ( y (,] +oo )  =  (
RR*  \  ( -oo [,] y ) ) ) )
5655rspcev 3085 . . . . . . . . . 10  |-  ( ( ( -oo [,] y
)  e.  ran  [,]  /\  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y
) ) )  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x ) )
5728, 53, 56syl2anc 661 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( y (,] +oo )  =  ( RR*  \  x
) )
58 xrex 11000 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4452 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 5 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5102 . . . . . . . . 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 5878 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,] +oo ) ) : RR* --> ran  F
64 frn 5577 . . . . . . 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 2443 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( -oo [,) y ) )  =  ( y  e.  RR*  |->  ( -oo [,) y ) )
67 fnovrn 6250 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( y [,] +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1305 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,] +oo )  e. 
ran  [,] )
69 df-ico 11318 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9454 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 11143 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  < +oo ) )
72 xrltle 11138 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\ +oo  e.  RR* )  ->  (
z  < +oo  ->  z  <_ +oo ) )
73723adant2 1007 . . . . . . . . . . . . . . . 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 11144 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <_  z )  -> -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 11328 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( -oo [,] +oo )
)
78 uncom 3512 . . . . . . . . . . . . 13  |-  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( ( y [,] +oo )  u.  ( -oo [,) y ) )
7977, 78, 453eqtr3g 2498 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  u.  ( -oo [,) y
) )  =  RR* )
80 iccssxr 11390 . . . . . . . . . . . . 13  |-  ( y [,] +oo )  C_  RR*
81 incom 3555 . . . . . . . . . . . . . 14  |-  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  ( ( -oo [,) y
)  i^i  ( y [,] +oo ) )
8269, 35, 70ixxdisj 11327 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8326, 31, 82mp3an13 1305 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8481, 83syl5eq 2487 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  (/) )
85 uneqdifeq 3779 . . . . . . . . . . . . 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 663 . . . . . . . . . . . 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 2448 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) )
89 difeq2 3480 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] +oo )  ->  ( RR*  \  x )  =  (
RR*  \  ( y [,] +oo ) ) )
9089eqeq2d 2454 . . . . . . . . . . 11  |-  ( x  =  ( y [,] +oo )  ->  ( ( -oo [,) y )  =  ( RR*  \  x
)  <->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) ) )
9190rspcev 3085 . . . . . . . . . 10  |-  ( ( ( y [,] +oo )  e.  ran  [,]  /\  ( -oo [,) y )  =  ( RR*  \  (
y [,] +oo )
) )  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
9268, 88, 91syl2anc 661 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( -oo [,) y )  =  (
RR*  \  x )
)
936, 60elrnmpti 5102 . . . . . . . . 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 5878 . . . . . . 7  |-  ( y  e.  RR*  |->  ( -oo [,) y ) ) :
RR* --> ran  F
96 frn 5577 . . . . . . 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 3543 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) )  C_  ran  F
99 fiss 7686 . . . . 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 672 . . . 4  |-  ( fi
`  ( ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) )  C_  ( fi ` 
ran  F )
101 tgss 18585 . . . 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 672 . . 3  |-  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) )
1033, 102eqsstri 3398 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 18822 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 18608 . . 3  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ran  F  C_  (ordTop `  <_  ) )  ->  ( topGen `  ( fi ` 
ran  F ) ) 
C_  (ordTop `  <_  ) )
106104, 23, 105mp2an 672 . 2  |-  ( topGen `  ( fi `  ran  F ) )  C_  (ordTop ` 
<_  )
107103, 106eqssi 3384 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728   _Vcvv 2984    \ cdif 3337    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   ~Pcpw 3872   class class class wbr 4304    e. cmpt 4362    X. cxp 4850   ran crn 4853    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   ficfi 7672   +oocpnf 9427   -oocmnf 9428   RR*cxr 9429    < clt 9430    <_ cle 9431   (,]cioc 11313   [,)cico 11314   [,]cicc 11315   topGenctg 14388  ordTopcordt 14449   Topctop 18510   Clsdccld 18632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fi 7673  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-ioc 11317  df-ico 11318  df-icc 11319  df-topgen 14394  df-ordt 14451  df-ps 15382  df-tsr 15383  df-top 18515  df-bases 18517  df-topon 18518  df-cld 18635
This theorem is referenced by: (None)
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