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Theorem lecldbas 17275
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 2435 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  =  ran  ( y  e. 
RR*  |->  ( y (,] 
+oo ) )
2 eqid 2435 . . . 4  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  (  -oo [,) y ) )
31, 2leordtval2 17268 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) )
4 fvex 5734 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5734 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 10995 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5583 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 8 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6214 . . . . . . . . . 10  |-  ( [,] 
Fn  ( RR*  X.  RR* )  ->  ( x  e. 
ran  [,]  <->  E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b ) ) )
119, 10ax-mp 8 . . . . . . . . 9  |-  ( x  e.  ran  [,]  <->  E. a  e.  RR*  E. b  e. 
RR*  x  =  ( a [,] b ) )
12 difeq2 3451 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 17270 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 17263 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 17087 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2523 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2818 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2818 . . . . . . . . 9  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x
)  e.  (ordTop `  <_  ) )
2011, 19sylbi 188 . . . . . . . 8  |-  ( x  e.  ran  [,]  ->  (
RR*  \  x )  e.  (ordTop `  <_  ) )
216, 20fmpti 5884 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5589 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 8 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4340 . . . . 5  |-  ran  F  e.  _V
25 eqid 2435 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) )  =  ( y  e.  RR*  |->  ( y (,]  +oo ) )
26 mnfxr 10706 . . . . . . . . . . 11  |-  -oo  e.  RR*
27 fnovrn 6213 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  -oo  e.  RR*  /\  y  e. 
RR* )  ->  (  -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1269 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,] y )  e.  ran  [,] )
2926a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  e.  RR* )
30 id 20 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  e. 
RR* )
31 pnfxr 10705 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  +oo  e.  RR* )
33 mnfle 10721 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  <_  y )
34 pnfge 10719 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_  +oo )
35 df-icc 10915 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 10913 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9136 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 10740 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
39 xrlelttr 10738 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <  z ) )
40 xrltle 10734 . . . . . . . . . . . . . . . . 17  |-  ( ( 
-oo  e.  RR*  /\  z  e.  RR* )  ->  (  -oo  <  z  ->  -oo  <_  z ) )
41403adant2 976 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (  -oo  <  z  ->  -oo  <_  z ) )
4239, 41syld 42 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <_  z ) )
4335, 36, 37, 35, 38, 42ixxun 10924 . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  ( 
-oo [,]  +oo ) )
45 iccmax 10978 . . . . . . . . . . . . 13  |-  (  -oo [,] 
+oo )  =  RR*
4644, 45syl6eq 2483 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  RR* )
47 iccssxr 10985 . . . . . . . . . . . . 13  |-  (  -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 10923 . . . . . . . . . . . . . 14  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo [,] y )  i^i  ( y (,]  +oo ) )  =  (/) )
4926, 31, 48mp3an13 1270 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  i^i  ( y (,]  +oo ) )  =  (/) )
50 uneqdifeq 3708 . . . . . . . . . . . . 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 645 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( (  -oo [,] y
)  u.  ( y (,]  +oo ) )  = 
RR* 
<->  ( RR*  \  (  -oo [,] y ) )  =  ( y (,] 
+oo ) ) )
5246, 51mpbid 202 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  (  -oo [,] y
) )  =  ( y (,]  +oo )
)
5352eqcomd 2440 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) )
54 difeq2 3451 . . . . . . . . . . . 12  |-  ( x  =  (  -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  (  -oo [,] y ) ) )
5554eqeq2d 2446 . . . . . . . . . . 11  |-  ( x  =  (  -oo [,] y )  ->  (
( y (,]  +oo )  =  ( RR*  \  x )  <->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) ) )
5655rspcev 3044 . . . . . . . . . 10  |-  ( ( (  -oo [,] y
)  e.  ran  [,]  /\  ( y (,]  +oo )  =  ( RR*  \  (  -oo [,] y
) ) )  ->  E. x  e.  ran  [,] ( y (,]  +oo )  =  ( RR*  \  x ) )
5728, 53, 56syl2anc 643 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( y (,]  +oo )  =  (
RR*  \  x )
)
58 xrex 10601 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4343 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 8 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5113 . . . . . . . . 9  |-  ( ( y (,]  +oo )  e.  ran  F  <->  E. x  e.  ran  [,] ( y (,]  +oo )  =  (
RR*  \  x )
)
6257, 61sylibr 204 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  e.  ran  F )
6325, 62fmpti 5884 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F
64 frn 5589 . . . . . . 7  |-  ( ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F  ->  ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  C_  ran  F )
6563, 64ax-mp 8 . . . . . 6  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  C_  ran  F
66 eqid 2435 . . . . . . . 8  |-  ( y  e.  RR*  |->  (  -oo [,) y ) )  =  ( y  e.  RR*  |->  (  -oo [,) y ) )
67 fnovrn 6213 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( y [,]  +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1270 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,]  +oo )  e.  ran  [,] )
69 df-ico 10914 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9135 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 10739 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <  +oo ) )
72 xrltle 10734 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\  +oo  e.  RR* )  ->  (
z  <  +oo  ->  z  <_  +oo ) )
73723adant2 976 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( z  <  +oo  ->  z  <_  +oo ) )
7471, 73syld 42 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
75 xrletr 10740 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <_  z )  ->  -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 10924 . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( 
-oo [,]  +oo ) )
78 uncom 3483 . . . . . . . . . . . . 13  |-  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( ( y [,]  +oo )  u.  (  -oo [,) y ) )
7977, 78, 453eqtr3g 2490 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  u.  (  -oo [,) y
) )  =  RR* )
80 iccssxr 10985 . . . . . . . . . . . . 13  |-  ( y [,]  +oo )  C_  RR*
81 incom 3525 . . . . . . . . . . . . . 14  |-  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  ( (  -oo [,) y
)  i^i  ( y [,]  +oo ) )
8269, 35, 70ixxdisj 10923 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo [,) y )  i^i  ( y [,]  +oo ) )  =  (/) )
8326, 31, 82mp3an13 1270 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( 
-oo [,) y )  i^i  ( y [,]  +oo ) )  =  (/) )
8481, 83syl5eq 2479 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  (/) )
85 uneqdifeq 3708 . . . . . . . . . . . . 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 645 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( y [,]  +oo )  u.  (  -oo [,) y ) )  = 
RR* 
<->  ( RR*  \  (
y [,]  +oo ) )  =  (  -oo [,) y ) ) )
8779, 86mpbid 202 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( y [,]  +oo ) )  =  ( 
-oo [,) y ) )
8887eqcomd 2440 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) )
89 difeq2 3451 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] 
+oo )  ->  ( RR*  \  x )  =  ( RR*  \  (
y [,]  +oo ) ) )
9089eqeq2d 2446 . . . . . . . . . . 11  |-  ( x  =  ( y [,] 
+oo )  ->  (
(  -oo [,) y )  =  ( RR*  \  x
)  <->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) ) )
9190rspcev 3044 . . . . . . . . . 10  |-  ( ( ( y [,]  +oo )  e.  ran  [,]  /\  (  -oo [,) y )  =  ( RR*  \  (
y [,]  +oo ) ) )  ->  E. x  e.  ran  [,] (  -oo [,) y )  =  (
RR*  \  x )
)
9268, 88, 91syl2anc 643 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] (  -oo [,) y )  =  (
RR*  \  x )
)
936, 60elrnmpti 5113 . . . . . . . . 9  |-  ( ( 
-oo [,) y )  e. 
ran  F  <->  E. x  e.  ran  [,] (  -oo [,) y
)  =  ( RR*  \  x ) )
9492, 93sylibr 204 . . . . . . . 8  |-  ( y  e.  RR*  ->  (  -oo [,) y )  e.  ran  F )
9566, 94fmpti 5884 . . . . . . 7  |-  ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F
96 frn 5589 . . . . . . 7  |-  ( ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F  ->  ran  ( y  e.  RR*  |->  (  -oo [,) y ) )  C_  ran  F )
9795, 96ax-mp 8 . . . . . 6  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  C_  ran  F
9865, 97unssi 3514 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) )  C_  ran  F
99 fiss 7421 . . . . 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 654 . . . 4  |-  ( fi
`  ( ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  u. 
ran  ( y  e. 
RR*  |->  (  -oo [,) y ) ) ) )  C_  ( fi ` 
ran  F )
101 tgss 17025 . . . 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 654 . . 3  |-  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) )
1033, 102eqsstri 3370 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 17262 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 17048 . . 3  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ran  F  C_  (ordTop `  <_  ) )  ->  ( topGen `  ( fi ` 
ran  F ) ) 
C_  (ordTop `  <_  ) )
106104, 23, 105mp2an 654 . 2  |-  ( topGen `  ( fi `  ran  F ) )  C_  (ordTop ` 
<_  )
107103, 106eqssi 3356 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   ficfi 7407    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   (,]cioc 10909   [,)cico 10910   [,]cicc 10911   topGenctg 13657  ordTopcordt 13713   Topctop 16950   Clsdccld 17072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-ioc 10913  df-ico 10914  df-icc 10915  df-topgen 13659  df-ordt 13717  df-ps 14621  df-tsr 14622  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075
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