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Theorem lecldbas 17199
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 2381 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  =  ran  ( y  e. 
RR*  |->  ( y (,] 
+oo ) )
2 eqid 2381 . . . 4  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  (  -oo [,) y ) )
31, 2leordtval2 17192 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) )
4 fvex 5676 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5676 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 10929 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5525 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 8 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6155 . . . . . . . . . 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 3396 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 17194 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 17187 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 17012 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2469 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2763 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2763 . . . . . . . . 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 5825 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5531 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 8 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4283 . . . . 5  |-  ran  F  e.  _V
25 eqid 2381 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) )  =  ( y  e.  RR*  |->  ( y (,]  +oo ) )
26 mnfxr 10640 . . . . . . . . . . 11  |-  -oo  e.  RR*
27 fnovrn 6154 . . . . . . . . . . 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 10639 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  +oo  e.  RR* )
33 mnfle 10655 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  <_  y )
34 pnfge 10653 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_  +oo )
35 df-icc 10849 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 10847 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9071 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 10674 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
39 xrlelttr 10672 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <  z ) )
40 xrltle 10668 . . . . . . . . . . . . . . . . 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 10858 . . . . . . . . . . . . . 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 10912 . . . . . . . . . . . . 13  |-  (  -oo [,] 
+oo )  =  RR*
4644, 45syl6eq 2429 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  RR* )
47 iccssxr 10919 . . . . . . . . . . . . 13  |-  (  -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 10857 . . . . . . . . . . . . . 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 3653 . . . . . . . . . . . . 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 2386 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) )
54 difeq2 3396 . . . . . . . . . . . 12  |-  ( x  =  (  -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  (  -oo [,] y ) ) )
5554eqeq2d 2392 . . . . . . . . . . 11  |-  ( x  =  (  -oo [,] y )  ->  (
( y (,]  +oo )  =  ( RR*  \  x )  <->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) ) )
5655rspcev 2989 . . . . . . . . . 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 10535 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4286 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 8 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5055 . . . . . . . . 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 5825 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F
64 frn 5531 . . . . . . 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 2381 . . . . . . . 8  |-  ( y  e.  RR*  |->  (  -oo [,) y ) )  =  ( y  e.  RR*  |->  (  -oo [,) y ) )
67 fnovrn 6154 . . . . . . . . . . 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 10848 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9070 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 10673 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <  +oo ) )
72 xrltle 10668 . . . . . . . . . . . . . . . . 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 10674 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <_  z )  ->  -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 10858 . . . . . . . . . . . . . 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 3428 . . . . . . . . . . . . 13  |-  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( ( y [,]  +oo )  u.  (  -oo [,) y ) )
7977, 78, 453eqtr3g 2436 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  u.  (  -oo [,) y
) )  =  RR* )
80 iccssxr 10919 . . . . . . . . . . . . 13  |-  ( y [,]  +oo )  C_  RR*
81 incom 3470 . . . . . . . . . . . . . 14  |-  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  ( (  -oo [,) y
)  i^i  ( y [,]  +oo ) )
8269, 35, 70ixxdisj 10857 . . . . . . . . . . . . . . 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 2425 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  (/) )
85 uneqdifeq 3653 . . . . . . . . . . . . 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 2386 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) )
89 difeq2 3396 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] 
+oo )  ->  ( RR*  \  x )  =  ( RR*  \  (
y [,]  +oo ) ) )
9089eqeq2d 2392 . . . . . . . . . . 11  |-  ( x  =  ( y [,] 
+oo )  ->  (
(  -oo [,) y )  =  ( RR*  \  x
)  <->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) ) )
9190rspcev 2989 . . . . . . . . . 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 5055 . . . . . . . . 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 5825 . . . . . . 7  |-  ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F
96 frn 5531 . . . . . . 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 3459 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) )  C_  ran  F
99 fiss 7358 . . . . 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 16950 . . . 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 3315 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 17186 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 16973 . . 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 3301 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 1717   E.wrex 2644   _Vcvv 2893    \ cdif 3254    u. cun 3255    i^i cin 3256    C_ wss 3257   (/)c0 3565   ~Pcpw 3736   class class class wbr 4147    e. cmpt 4201    X. cxp 4810   ran crn 4813    Fn wfn 5383   -->wf 5384   ` cfv 5388  (class class class)co 6014   ficfi 7344    +oocpnf 9044    -oocmnf 9045   RR*cxr 9046    < clt 9047    <_ cle 9048   (,]cioc 10843   [,)cico 10844   [,]cicc 10845   topGenctg 13586  ordTopcordt 13642   Topctop 16875   Clsdccld 16997
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-fi 7345  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-ioc 10847  df-ico 10848  df-icc 10849  df-topgen 13588  df-ordt 13646  df-ps 14550  df-tsr 14551  df-top 16880  df-bases 16882  df-topon 16883  df-cld 17000
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