MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lecldbas Unicode version

Theorem lecldbas 16965
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 2296 . . . 4  |-  ran  (
y  e.  RR*  |->  ( y (,]  +oo ) )  =  ran  ( y  e. 
RR*  |->  ( y (,] 
+oo ) )
2 eqid 2296 . . . 4  |-  ran  (
y  e.  RR*  |->  (  -oo [,) y ) )  =  ran  ( y  e. 
RR*  |->  (  -oo [,) y ) )
31, 2leordtval2 16958 . . 3  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) ) ) )
4 fvex 5555 . . . 4  |-  ( fi
`  ran  F )  e.  _V
5 fvex 5555 . . . . . 6  |-  (ordTop `  <_  )  e.  _V
6 lecldbas.1 . . . . . . . 8  |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x
) )
7 iccf 10758 . . . . . . . . . . 11  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
8 ffn 5405 . . . . . . . . . . 11  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
97, 8ax-mp 8 . . . . . . . . . 10  |-  [,]  Fn  ( RR*  X.  RR* )
10 ovelrn 6012 . . . . . . . . . 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 3301 . . . . . . . . . . . 12  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  =  ( RR*  \  (
a [,] b ) ) )
13 iccordt 16960 . . . . . . . . . . . . 13  |-  ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )
14 letopuni 16953 . . . . . . . . . . . . . 14  |-  RR*  =  U. (ordTop `  <_  )
1514cldopn 16784 . . . . . . . . . . . . 13  |-  ( ( a [,] b )  e.  ( Clsd `  (ordTop ` 
<_  ) )  ->  ( RR*  \  ( a [,] b ) )  e.  (ordTop `  <_  ) )
1613, 15ax-mp 8 . . . . . . . . . . . 12  |-  ( RR*  \  ( a [,] b
) )  e.  (ordTop `  <_  )
1712, 16syl6eqel 2384 . . . . . . . . . . 11  |-  ( x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1817rexlimivw 2676 . . . . . . . . . 10  |-  ( E. b  e.  RR*  x  =  ( a [,] b )  ->  ( RR*  \  x )  e.  (ordTop `  <_  ) )
1918rexlimivw 2676 . . . . . . . . 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 5699 . . . . . . 7  |-  F : ran  [,] --> (ordTop `  <_  )
22 frn 5411 . . . . . . 7  |-  ( F : ran  [,] --> (ordTop `  <_  )  ->  ran  F  C_  (ordTop `  <_  ) )
2321, 22ax-mp 8 . . . . . 6  |-  ran  F  C_  (ordTop `  <_  )
245, 23ssexi 4175 . . . . 5  |-  ran  F  e.  _V
25 eqid 2296 . . . . . . . 8  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) )  =  ( y  e.  RR*  |->  ( y (,]  +oo ) )
26 mnfxr 10472 . . . . . . . . . . 11  |-  -oo  e.  RR*
27 fnovrn 6011 . . . . . . . . . . 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 10471 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
3231a1i 10 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  +oo  e.  RR* )
33 mnfle 10486 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  -oo  <_  y )
34 pnfge 10485 . . . . . . . . . . . . . 14  |-  ( y  e.  RR*  ->  y  <_  +oo )
35 df-icc 10679 . . . . . . . . . . . . . . 15  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <_  b ) } )
36 df-ioc 10677 . . . . . . . . . . . . . . 15  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <  c  /\  c  <_  b ) } )
37 xrltnle 8907 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <  z  <->  -.  z  <_  y ) )
38 xrletr 10505 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <_  y  /\  y  <_  +oo )  ->  z  <_  +oo ) )
39 xrlelttr 10503 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <  z )  ->  -oo  <  z ) )
40 xrltle 10499 . . . . . . . . . . . . . . . . 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 10688 . . . . . . . . . . . . . 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 10741 . . . . . . . . . . . . 13  |-  (  -oo [,] 
+oo )  =  RR*
4644, 45syl6eq 2344 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( 
-oo [,] y )  u.  ( y (,]  +oo ) )  =  RR* )
47 iccssxr 10748 . . . . . . . . . . . . 13  |-  (  -oo [,] y )  C_  RR*
4835, 36, 37ixxdisj 10687 . . . . . . . . . . . . . 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 3555 . . . . . . . . . . . . 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 2301 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) )
54 difeq2 3301 . . . . . . . . . . . 12  |-  ( x  =  (  -oo [,] y )  ->  ( RR*  \  x )  =  ( RR*  \  (  -oo [,] y ) ) )
5554eqeq2d 2307 . . . . . . . . . . 11  |-  ( x  =  (  -oo [,] y )  ->  (
( y (,]  +oo )  =  ( RR*  \  x )  <->  ( y (,]  +oo )  =  (
RR*  \  (  -oo [,] y ) ) ) )
5655rspcev 2897 . . . . . . . . . 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 10367 . . . . . . . . . . 11  |-  RR*  e.  _V
59 difexg 4178 . . . . . . . . . . 11  |-  ( RR*  e.  _V  ->  ( RR*  \  x )  e.  _V )
6058, 59ax-mp 8 . . . . . . . . . 10  |-  ( RR*  \  x )  e.  _V
616, 60elrnmpti 4946 . . . . . . . . 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 5699 . . . . . . 7  |-  ( y  e.  RR*  |->  ( y (,]  +oo ) ) :
RR* --> ran  F
64 frn 5411 . . . . . . 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 2296 . . . . . . . 8  |-  ( y  e.  RR*  |->  (  -oo [,) y ) )  =  ( y  e.  RR*  |->  (  -oo [,) y ) )
67 fnovrn 6011 . . . . . . . . . . 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 10678 . . . . . . . . . . . . . . 15  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { c  e.  RR*  |  (
a  <_  c  /\  c  <  b ) } )
70 xrlenlt 8906 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  z  e.  RR* )  ->  (
y  <_  z  <->  -.  z  <  y ) )
71 xrltletr 10504 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
z  <  y  /\  y  <_  +oo )  ->  z  <  +oo ) )
72 xrltle 10499 . . . . . . . . . . . . . . . . 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 10505 . . . . . . . . . . . . . . 15  |-  ( ( 
-oo  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
(  -oo  <_  y  /\  y  <_  z )  ->  -oo  <_  z ) )
7669, 35, 70, 35, 74, 75ixxun 10688 . . . . . . . . . . . . . 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 3332 . . . . . . . . . . . . 13  |-  ( ( 
-oo [,) y )  u.  ( y [,]  +oo ) )  =  ( ( y [,]  +oo )  u.  (  -oo [,) y ) )
7977, 78, 453eqtr3g 2351 . . . . . . . . . . . 12  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  u.  (  -oo [,) y
) )  =  RR* )
80 iccssxr 10748 . . . . . . . . . . . . 13  |-  ( y [,]  +oo )  C_  RR*
81 incom 3374 . . . . . . . . . . . . . 14  |-  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  ( (  -oo [,) y
)  i^i  ( y [,]  +oo ) )
8269, 35, 70ixxdisj 10687 . . . . . . . . . . . . . . 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 2340 . . . . . . . . . . . . 13  |-  ( y  e.  RR*  ->  ( ( y [,]  +oo )  i^i  (  -oo [,) y
) )  =  (/) )
85 uneqdifeq 3555 . . . . . . . . . . . . 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 2301 . . . . . . . . . 10  |-  ( y  e.  RR*  ->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) )
89 difeq2 3301 . . . . . . . . . . . 12  |-  ( x  =  ( y [,] 
+oo )  ->  ( RR*  \  x )  =  ( RR*  \  (
y [,]  +oo ) ) )
9089eqeq2d 2307 . . . . . . . . . . 11  |-  ( x  =  ( y [,] 
+oo )  ->  (
(  -oo [,) y )  =  ( RR*  \  x
)  <->  (  -oo [,) y )  =  (
RR*  \  ( y [,]  +oo ) ) ) )
9190rspcev 2897 . . . . . . . . . 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 4946 . . . . . . . . 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 5699 . . . . . . 7  |-  ( y  e.  RR*  |->  (  -oo [,) y ) ) :
RR* --> ran  F
96 frn 5411 . . . . . . 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 3363 . . . . 5  |-  ( ran  ( y  e.  RR*  |->  ( y (,]  +oo ) )  u.  ran  ( y  e.  RR*  |->  (  -oo [,) y ) ) )  C_  ran  F
99 fiss 7193 . . . . 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 16722 . . . 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 3221 . 2  |-  (ordTop `  <_  )  C_  ( topGen `  ( fi `  ran  F ) )
104 letop 16952 . . 3  |-  (ordTop `  <_  )  e.  Top
105 tgfiss 16745 . . 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 3208 1  |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   ficfi 7180    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   (,]cioc 10673   [,)cico 10674   [,]cicc 10675   topGenctg 13358  ordTopcordt 13414   Topctop 16647   Clsdccld 16769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ioc 10677  df-ico 10678  df-icc 10679  df-topgen 13360  df-ordt 13418  df-ps 14322  df-tsr 14323  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772
  Copyright terms: Public domain W3C validator