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Theorem lecldbas 19502
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 2467 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,] +oo ) )  =  ran  ( y  e.  RR*  |->  ( y (,] +oo ) )
2 eqid 2467 . . . 4  |-  ran  (
y  e.  RR*  |->  ( -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  ( -oo [,) y ) )
31, 2leordtval2 19495 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) ) ) )
4 fvex 5875 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5875 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 11622 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5730 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 5 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6434 . . . . . . . . . 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 3616 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 19497 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 19490 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 19314 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2563 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2952 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2952 . . . . . . . . 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 6043 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5736 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 5 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4592 . . . . 5  |-  ran  F  e.  _V
25 eqid 2467 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,] +oo ) )  =  ( y  e. 
RR*  |->  ( y (,] +oo ) )
26 mnfxr 11322 . . . . . . . . . . 11  |- -oo  e.  RR*
27 fnovrn 6433 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\  y  e. 
RR* )  ->  ( -oo [,] y )  e. 
ran  [,] )
289, 26, 27mp3an12 1314 . . . . . . . . . 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 11320 . . . . . . . . . . . . . . 15  |- +oo  e.  RR*
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> +oo  e.  RR* )
33 mnfle 11341 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  -> -oo  <_  y )
34 pnfge 11338 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_ +oo )
35 df-icc 11535 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 11533 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 9652 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 11360 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_ +oo )  ->  z  <_ +oo ) )
39 xrlelttr 11358 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <  z )  -> -oo  <  z ) )
40 xrltle 11354 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  e.  RR*  /\  z  e.  RR* )  ->  ( -oo  <  z  -> -oo  <_  z ) )
41403adant2 1015 . . . . . . . . . . . . . . . 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 11544 . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  ( -oo [,] +oo )
)
45 iccmax 11599 . . . . . . . . . . . . 13  |-  ( -oo [,] +oo )  =  RR*
4644, 45syl6eq 2524 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  u.  ( y (,] +oo ) )  =  RR* )
47 iccssxr 11606 . . . . . . . . . . . . 13  |-  ( -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 11543 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
4926, 31, 48mp3an13 1315 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,] y )  i^i  ( y (,] +oo ) )  =  (/) )
50 uneqdifeq 3915 . . . . . . . . . . . . 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 2475 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,] +oo )  =  ( RR*  \  ( -oo [,] y ) ) )
54 difeq2 3616 . . . . . . . . . . . 12  |-  ( x  =  ( -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  ( -oo [,] y ) ) )
5554eqeq2d 2481 . . . . . . . . . . 11  |-  ( x  =  ( -oo [,] y )  ->  (
( y (,] +oo )  =  ( RR*  \  x )  <->  ( y (,] +oo )  =  (
RR*  \  ( -oo [,] y ) ) ) )
5655rspcev 3214 . . . . . . . . . 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 11216 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4595 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 5 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 5252 . . . . . . . . 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 6043 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,] +oo ) ) : RR* --> ran  F
64 frn 5736 . . . . . . 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 2467 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( -oo [,) y ) )  =  ( y  e.  RR*  |->  ( -oo [,) y ) )
67 fnovrn 6433 . . . . . . . . . . 11  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( y [,] +oo )  e.  ran  [,] )
689, 31, 67mp3an13 1315 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y [,] +oo )  e. 
ran  [,] )
69 df-ico 11534 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 9651 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 11359 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_ +oo )  ->  z  < +oo ) )
72 xrltle 11354 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR*  /\ +oo  e.  RR* )  ->  (
z  < +oo  ->  z  <_ +oo ) )
73723adant2 1015 . . . . . . . . . . . . . . . 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 11360 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( -oo  <_  y  /\  y  <_  z )  -> -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 11544 . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( -oo [,] +oo )
)
78 uncom 3648 . . . . . . . . . . . . 13  |-  ( ( -oo [,) y )  u.  ( y [,] +oo ) )  =  ( ( y [,] +oo )  u.  ( -oo [,) y ) )
7977, 78, 453eqtr3g 2531 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  u.  ( -oo [,) y
) )  =  RR* )
80 iccssxr 11606 . . . . . . . . . . . . 13  |-  ( y [,] +oo )  C_  RR*
81 incom 3691 . . . . . . . . . . . . . 14  |-  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  ( ( -oo [,) y
)  i^i  ( y [,] +oo ) )
8269, 35, 70ixxdisj 11543 . . . . . . . . . . . . . . 15  |-  ( ( -oo  e.  RR*  /\  y  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8326, 31, 82mp3an13 1315 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  ( ( -oo [,) y )  i^i  ( y [,] +oo ) )  =  (/) )
8481, 83syl5eq 2520 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,] +oo )  i^i  ( -oo [,) y
) )  =  (/) )
85 uneqdifeq 3915 . . . . . . . . . . . . 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 2475 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) )
89 difeq2 3616 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] +oo )  ->  ( RR*  \  x )  =  (
RR*  \  ( y [,] +oo ) ) )
9089eqeq2d 2481 . . . . . . . . . . 11  |-  ( x  =  ( y [,] +oo )  ->  ( ( -oo [,) y )  =  ( RR*  \  x
)  <->  ( -oo [,) y )  =  (
RR*  \  ( y [,] +oo ) ) ) )
9190rspcev 3214 . . . . . . . . . 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 5252 . . . . . . . . 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 6043 . . . . . . 7  |-  ( y  e.  RR*  |->  ( -oo [,) y ) ) :
RR* --> ran  F
96 frn 5736 . . . . . . 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 3679 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,] +oo ) )  u.  ran  ( y  e.  RR*  |->  ( -oo [,) y ) ) )  C_  ran  F
99 fiss 7883 . . . . 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 19252 . . . 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 3534 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 19489 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 19275 . . 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 3520 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   ficfi 7869   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626    < clt 9627    <_ cle 9628   (,]cioc 11529   [,)cico 11530   [,]cicc 11531   topGenctg 14692  ordTopcordt 14753   Topctop 19177   Clsdccld 19299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7870  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-ioc 11533  df-ico 11534  df-icc 11535  df-topgen 14698  df-ordt 14755  df-ps 15686  df-tsr 15687  df-top 19182  df-bases 19184  df-topon 19185  df-cld 19302
This theorem is referenced by: (None)
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