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Theorem lecldbas 17237
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 2404 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  =  ran  ( y  e. 
RR*  |->  ( y (,] 
+oo ) )
2 eqid 2404 . . . 4  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  (  -oo [,) y ) )
31, 2leordtval2 17230 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) )
4 fvex 5701 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5701 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 10959 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5550 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 8 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6181 . . . . . . . . . 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 3419 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 17232 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 17225 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 17050 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2492 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2786 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2786 . . . . . . . . 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 5851 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5556 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 8 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4308 . . . . 5  |-  ran  F  e.  _V
25 eqid 2404 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) )  =  ( y  e.  RR*  |->  ( y (,]  +oo ) )
26 mnfxr 10670 . . . . . . . . . . 11  |-  -oo  e.  RR*
27 fnovrn 6180 . . . . . . . . . . 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 10669 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  +oo  e.  RR* )
33 mnfle 10685 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  <_  y )
34 pnfge 10683 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_  +oo )
35 df-icc 10879 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 10877 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9100 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 10704 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
39 xrlelttr 10702 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <  z ) )
40 xrltle 10698 . . . . . . . . . . . . . . . . 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 10888 . . . . . . . . . . . . . 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 10942 . . . . . . . . . . . . 13  |-  (  -oo [,] 
+oo )  =  RR*
4644, 45syl6eq 2452 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  RR* )
47 iccssxr 10949 . . . . . . . . . . . . 13  |-  (  -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 10887 . . . . . . . . . . . . . 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 3676 . . . . . . . . . . . . 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 2409 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) )
54 difeq2 3419 . . . . . . . . . . . 12  |-  ( x  =  (  -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  (  -oo [,] y ) ) )
5554eqeq2d 2415 . . . . . . . . . . 11  |-  ( x  =  (  -oo [,] y )  ->  (
( y (,]  +oo )  =  ( RR*  \  x )  <->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) ) )
5655rspcev 3012 . . . . . . . . . 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 10565 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4311 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 8 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5080 . . . . . . . . 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 5851 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F
64 frn 5556 . . . . . . 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 2404 . . . . . . . 8  |-  ( y  e.  RR*  |->  (  -oo [,) y ) )  =  ( y  e.  RR*  |->  (  -oo [,) y ) )
67 fnovrn 6180 . . . . . . . . . . 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 10878 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9099 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 10703 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <  +oo ) )
72 xrltle 10698 . . . . . . . . . . . . . . . . 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 10704 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <_  z )  ->  -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 10888 . . . . . . . . . . . . . 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 3451 . . . . . . . . . . . . 13  |-  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( ( y [,]  +oo )  u.  (  -oo [,) y ) )
7977, 78, 453eqtr3g 2459 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  u.  (  -oo [,) y
) )  =  RR* )
80 iccssxr 10949 . . . . . . . . . . . . 13  |-  ( y [,]  +oo )  C_  RR*
81 incom 3493 . . . . . . . . . . . . . 14  |-  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  ( (  -oo [,) y
)  i^i  ( y [,]  +oo ) )
8269, 35, 70ixxdisj 10887 . . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  (/) )
85 uneqdifeq 3676 . . . . . . . . . . . . 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 2409 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) )
89 difeq2 3419 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] 
+oo )  ->  ( RR*  \  x )  =  ( RR*  \  (
y [,]  +oo ) ) )
9089eqeq2d 2415 . . . . . . . . . . 11  |-  ( x  =  ( y [,] 
+oo )  ->  (
(  -oo [,) y )  =  ( RR*  \  x
)  <->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) ) )
9190rspcev 3012 . . . . . . . . . 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 5080 . . . . . . . . 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 5851 . . . . . . 7  |-  ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F
96 frn 5556 . . . . . . 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 3482 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) )  C_  ran  F
99 fiss 7387 . . . . 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 16988 . . . 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 3338 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 17224 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 17011 . . 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 3324 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   ficfi 7373    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   (,]cioc 10873   [,)cico 10874   [,]cicc 10875   topGenctg 13620  ordTopcordt 13676   Topctop 16913   Clsdccld 17035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-ioc 10877  df-ico 10878  df-icc 10879  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038
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