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Theorem lecldbas 16949
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 2283 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  =  ran  ( y  e. 
RR*  |->  ( y (,] 
+oo ) )
2 eqid 2283 . . . 4  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  (  -oo [,) y ) )
31, 2leordtval2 16942 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) )
4 fvex 5539 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5539 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 10742 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5389 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 8 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 5996 . . . . . . . . . 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 3288 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 16944 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 16937 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 16768 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2371 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2663 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2663 . . . . . . . . 9  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x
)  e.  (ordTop `  <_  ) )
2011, 19sylbi 187 . . . . . . . 8  |-  ( x  e.  ran  [,]  ->  (
RR*  \  x )  e.  (ordTop `  <_  ) )
216, 20fmpti 5683 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5395 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 8 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4159 . . . . 5  |-  ran  F  e.  _V
25 eqid 2283 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) )  =  ( y  e.  RR*  |->  ( y (,]  +oo ) )
26 mnfxr 10456 . . . . . . . . . . 11  |-  -oo  e.  RR*
27 fnovrn 5995 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  -oo  e.  RR*  /\  y  e. 
RR* )  ->  (  -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1267 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,] y )  e.  ran  [,] )
2926a1i 10 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  e.  RR* )
30 id 19 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  e. 
RR* )
31 pnfxr 10455 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
3231a1i 10 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  +oo  e.  RR* )
33 mnfle 10470 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  <_  y )
34 pnfge 10469 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_  +oo )
35 df-icc 10663 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 10661 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 8891 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 10489 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
39 xrlelttr 10487 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <  z ) )
40 xrltle 10483 . . . . . . . . . . . . . . . . 17  |-  ( ( 
-oo  e.  RR*  /\  z  e.  RR* )  ->  (  -oo  <  z  ->  -oo  <_  z ) )
41403adant2 974 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (  -oo  <  z  ->  -oo  <_  z ) )
4239, 41syld 40 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <_  z ) )
4335, 36, 37, 35, 38, 42ixxun 10672 . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  ( 
-oo [,]  +oo ) )
45 iccmax 10725 . . . . . . . . . . . . 13  |-  (  -oo [,] 
+oo )  =  RR*
4644, 45syl6eq 2331 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  RR* )
47 iccssxr 10732 . . . . . . . . . . . . 13  |-  (  -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 10671 . . . . . . . . . . . . . 14  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo [,] y )  i^i  ( y (,]  +oo ) )  =  (/) )
4926, 31, 48mp3an13 1268 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  i^i  ( y (,]  +oo ) )  =  (/) )
50 uneqdifeq 3542 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( (  -oo [,] y
)  u.  ( y (,]  +oo ) )  = 
RR* 
<->  ( RR*  \  (  -oo [,] y ) )  =  ( y (,] 
+oo ) ) )
5246, 51mpbid 201 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  (  -oo [,] y
) )  =  ( y (,]  +oo )
)
5352eqcomd 2288 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) )
54 difeq2 3288 . . . . . . . . . . . 12  |-  ( x  =  (  -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  (  -oo [,] y ) ) )
5554eqeq2d 2294 . . . . . . . . . . 11  |-  ( x  =  (  -oo [,] y )  ->  (
( y (,]  +oo )  =  ( RR*  \  x )  <->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) ) )
5655rspcev 2884 . . . . . . . . . 10  |-  ( ( (  -oo [,] y
)  e.  ran  [,]  /\  ( y (,]  +oo )  =  ( RR*  \  (  -oo [,] y
) ) )  ->  E. x  e.  ran  [,] ( y (,]  +oo )  =  ( RR*  \  x ) )
5728, 53, 56syl2anc 642 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] ( y (,]  +oo )  =  (
RR*  \  x )
)
58 xrex 10351 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4162 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 8 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 4930 . . . . . . . . 9  |-  ( ( y (,]  +oo )  e.  ran  F  <->  E. x  e.  ran  [,] ( y (,]  +oo )  =  (
RR*  \  x )
)
6257, 61sylibr 203 . . . . . . . 8  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  e.  ran  F )
6325, 62fmpti 5683 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F
64 frn 5395 . . . . . . 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 2283 . . . . . . . 8  |-  ( y  e.  RR*  |->  (  -oo [,) y ) )  =  ( y  e.  RR*  |->  (  -oo [,) y ) )
67 fnovrn 5995 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( y [,]  +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1268 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,]  +oo )  e.  ran  [,] )
69 df-ico 10662 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 8890 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 10488 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <  +oo ) )
72 xrltle 10483 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\  +oo  e.  RR* )  ->  (
z  <  +oo  ->  z  <_  +oo ) )
73723adant2 974 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( z  <  +oo  ->  z  <_  +oo ) )
7471, 73syld 40 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
75 xrletr 10489 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <_  z )  ->  -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 10672 . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( 
-oo [,]  +oo ) )
78 uncom 3319 . . . . . . . . . . . . 13  |-  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( ( y [,]  +oo )  u.  (  -oo [,) y ) )
7977, 78, 453eqtr3g 2338 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  u.  (  -oo [,) y
) )  =  RR* )
80 iccssxr 10732 . . . . . . . . . . . . 13  |-  ( y [,]  +oo )  C_  RR*
81 incom 3361 . . . . . . . . . . . . . 14  |-  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  ( (  -oo [,) y
)  i^i  ( y [,]  +oo ) )
8269, 35, 70ixxdisj 10671 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo [,) y )  i^i  ( y [,]  +oo ) )  =  (/) )
8326, 31, 82mp3an13 1268 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( 
-oo [,) y )  i^i  ( y [,]  +oo ) )  =  (/) )
8481, 83syl5eq 2327 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  (/) )
85 uneqdifeq 3542 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( ( y [,]  +oo )  u.  (  -oo [,) y ) )  = 
RR* 
<->  ( RR*  \  (
y [,]  +oo ) )  =  (  -oo [,) y ) ) )
8779, 86mpbid 201 . . . . . . . . . . 11  |-  ( y  e.  RR*  ->  ( RR*  \  ( y [,]  +oo ) )  =  ( 
-oo [,) y ) )
8887eqcomd 2288 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) )
89 difeq2 3288 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] 
+oo )  ->  ( RR*  \  x )  =  ( RR*  \  (
y [,]  +oo ) ) )
9089eqeq2d 2294 . . . . . . . . . . 11  |-  ( x  =  ( y [,] 
+oo )  ->  (
(  -oo [,) y )  =  ( RR*  \  x
)  <->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) ) )
9190rspcev 2884 . . . . . . . . . 10  |-  ( ( ( y [,]  +oo )  e.  ran  [,]  /\  (  -oo [,) y )  =  ( RR*  \  (
y [,]  +oo ) ) )  ->  E. x  e.  ran  [,] (  -oo [,) y )  =  (
RR*  \  x )
)
9268, 88, 91syl2anc 642 . . . . . . . . 9  |-  ( y  e.  RR*  ->  E. x  e.  ran  [,] (  -oo [,) y )  =  (
RR*  \  x )
)
936, 60elrnmpti 4930 . . . . . . . . 9  |-  ( ( 
-oo [,) y )  e. 
ran  F  <->  E. x  e.  ran  [,] (  -oo [,) y
)  =  ( RR*  \  x ) )
9492, 93sylibr 203 . . . . . . . 8  |-  ( y  e.  RR*  ->  (  -oo [,) y )  e.  ran  F )
9566, 94fmpti 5683 . . . . . . 7  |-  ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F
96 frn 5395 . . . . . . 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 3350 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) )  C_  ran  F
99 fiss 7177 . . . . 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 653 . . . 4  |-  ( fi
`  ( ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  u. 
ran  ( y  e. 
RR*  |->  (  -oo [,) y ) ) ) )  C_  ( fi ` 
ran  F )
101 tgss 16706 . . . 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 653 . . 3  |-  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) ) 
C_  ( topGen `  ( fi `  ran  F ) )
1033, 102eqsstri 3208 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 16936 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 16729 . . 3  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ran  F  C_  (ordTop `  <_  ) )  ->  ( topGen `  ( fi ` 
ran  F ) ) 
C_  (ordTop `  <_  ) )
106104, 23, 105mp2an 653 . 2  |-  ( topGen `  ( fi `  ran  F ) )  C_  (ordTop ` 
<_  )
107103, 106eqssi 3195 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   ficfi 7164    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,]cioc 10657   [,)cico 10658   [,]cicc 10659   topGenctg 13342  ordTopcordt 13398   Topctop 16631   Clsdccld 16753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ioc 10661  df-ico 10662  df-icc 10663  df-topgen 13344  df-ordt 13402  df-ps 14306  df-tsr 14307  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756
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