Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem9 Structured version   Unicode version

Theorem erdszelem9 28818
Description: Lemma for erdsze 28821. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
Assertion
Ref Expression
erdszelem9  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Distinct variable groups:    x, y, n, F    n, I, x, y    n, J, x, y    n, N, x, y    ph, n, x, y
Allowed substitution hints:    T( x, y, n)

Proof of Theorem erdszelem9
Dummy variables  w  z  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . . . . . 6  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 erdszelem.i . . . . . 6  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
4 ltso 9682 . . . . . 6  |-  <  Or  RR
51, 2, 3, 4erdszelem6 28815 . . . . 5  |-  ( ph  ->  I : ( 1 ... N ) --> NN )
65ffvelrnda 6032 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
I `  n )  e.  NN )
7 erdszelem.j . . . . . 6  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
8 gtso 9683 . . . . . 6  |-  `'  <  Or  RR
91, 2, 7, 8erdszelem6 28815 . . . . 5  |-  ( ph  ->  J : ( 1 ... N ) --> NN )
109ffvelrnda 6032 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( J `  n )  e.  NN )
11 opelxpi 5040 . . . 4  |-  ( ( ( I `  n
)  e.  NN  /\  ( J `  n )  e.  NN )  ->  <. ( I `  n
) ,  ( J `
 n ) >.  e.  ( NN  X.  NN ) )
126, 10, 11syl2anc 661 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  <. (
I `  n ) ,  ( J `  n ) >.  e.  ( NN  X.  NN ) )
13 erdszelem.t . . 3  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
1412, 13fmptd 6056 . 2  |-  ( ph  ->  T : ( 1 ... N ) --> ( NN  X.  NN ) )
15 fveq2 5872 . . . . . 6  |-  ( a  =  z  ->  ( T `  a )  =  ( T `  z ) )
16 fveq2 5872 . . . . . 6  |-  ( b  =  w  ->  ( T `  b )  =  ( T `  w ) )
1715, 16eqeqan12d 2480 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
18 eqeq12 2476 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( a  =  b  <-> 
z  =  w ) )
1917, 18imbi12d 320 . . . 4  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
20 fveq2 5872 . . . . . . 7  |-  ( a  =  w  ->  ( T `  a )  =  ( T `  w ) )
21 fveq2 5872 . . . . . . 7  |-  ( b  =  z  ->  ( T `  b )  =  ( T `  z ) )
2220, 21eqeqan12d 2480 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  w
)  =  ( T `
 z ) ) )
23 eqcom 2466 . . . . . 6  |-  ( ( T `  w )  =  ( T `  z )  <->  ( T `  z )  =  ( T `  w ) )
2422, 23syl6bb 261 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
25 eqeq12 2476 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
w  =  z ) )
26 eqcom 2466 . . . . . 6  |-  ( w  =  z  <->  z  =  w )
2725, 26syl6bb 261 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
z  =  w ) )
2824, 27imbi12d 320 . . . 4  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
29 elfzelz 11713 . . . . . . 7  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3029zred 10990 . . . . . 6  |-  ( z  e.  ( 1 ... N )  ->  z  e.  RR )
3130ssriv 3503 . . . . 5  |-  ( 1 ... N )  C_  RR
3231a1i 11 . . . 4  |-  ( ph  ->  ( 1 ... N
)  C_  RR )
33 biidd 237 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( ( T `
 z )  =  ( T `  w
)  ->  z  =  w )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
34 simpr1 1002 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  ( 1 ... N
) )
35 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  z  ->  (
I `  n )  =  ( I `  z ) )
36 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  z  ->  ( J `  n )  =  ( J `  z ) )
3735, 36opeq12d 4227 . . . . . . . . 9  |-  ( n  =  z  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  z ) ,  ( J `  z ) >. )
38 opex 4720 . . . . . . . . 9  |-  <. (
I `  z ) ,  ( J `  z ) >.  e.  _V
3937, 13, 38fvmpt 5956 . . . . . . . 8  |-  ( z  e.  ( 1 ... N )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
4034, 39syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
41 simpr2 1003 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  ( 1 ... N
) )
42 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  w  ->  (
I `  n )  =  ( I `  w ) )
43 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  w  ->  ( J `  n )  =  ( J `  w ) )
4442, 43opeq12d 4227 . . . . . . . . 9  |-  ( n  =  w  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  w ) ,  ( J `  w ) >. )
45 opex 4720 . . . . . . . . 9  |-  <. (
I `  w ) ,  ( J `  w ) >.  e.  _V
4644, 13, 45fvmpt 5956 . . . . . . . 8  |-  ( w  e.  ( 1 ... N )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4741, 46syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4840, 47eqeq12d 2479 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  <->  <. ( I `
 z ) ,  ( J `  z
) >.  =  <. (
I `  w ) ,  ( J `  w ) >. )
)
49 fvex 5882 . . . . . . . 8  |-  ( I `
 z )  e. 
_V
50 fvex 5882 . . . . . . . 8  |-  ( J `
 z )  e. 
_V
5149, 50opth 4730 . . . . . . 7  |-  ( <.
( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>. 
<->  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) )
5234, 30syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  RR )
5331, 41sseldi 3497 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  RR )
54 simpr3 1004 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  <_  w )
5552, 53, 54leltned 9753 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  w  =/=  z ) )
562adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  F : ( 1 ... N ) -1-1-> RR )
57 f1fveq 6171 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( 1 ... N ) -1-1-> RR  /\  ( z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N ) ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
5856, 34, 41, 57syl12anc 1226 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
5958, 26syl6bbr 263 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  w  =  z ) )
6059necon3bid 2715 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =/=  ( F `
 w )  <->  w  =/=  z ) )
6155, 60bitr4d 256 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  ( F `  z )  =/=  ( F `  w )
) )
6261biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  =/=  ( F `  w )
)
63 f1f 5787 . . . . . . . . . . . . . . . 16  |-  ( F : ( 1 ... N ) -1-1-> RR  ->  F : ( 1 ... N ) --> RR )
642, 63syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 1 ... N ) --> RR )
6564ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) --> RR )
6634adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  e.  ( 1 ... N
) )
6765, 66ffvelrnd 6033 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  e.  RR )
6841adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  w  e.  ( 1 ... N
) )
6965, 68ffvelrnd 6033 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  w )  e.  RR )
7067, 69lttri2d 9741 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  =/=  ( F `  w
)  <->  ( ( F `
 z )  < 
( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) ) )
7162, 70mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  <  ( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) )
721ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  N  e.  NN )
732ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) -1-1-> RR )
74 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  <  w )
7572, 73, 3, 4, 66, 68, 74erdszelem8 28817 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
I `  z )  =  ( I `  w )  ->  -.  ( F `  z )  <  ( F `  w ) ) )
7672, 73, 7, 8, 66, 68, 74erdszelem8 28817 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( J `  z )  =  ( J `  w )  ->  -.  ( F `  z ) `'  <  ( F `  w ) ) )
7775, 76anim12d 563 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  -> 
( -.  ( F `
 z )  < 
( F `  w
)  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) ) )
78 ioran 490 . . . . . . . . . . . . 13  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  w )  <  ( F `  z ) ) )
79 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( F `
 z )  e. 
_V
80 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( F `
 w )  e. 
_V
8179, 80brcnv 5195 . . . . . . . . . . . . . . 15  |-  ( ( F `  z ) `'  <  ( F `  w )  <->  ( F `  w )  <  ( F `  z )
)
8281notbii 296 . . . . . . . . . . . . . 14  |-  ( -.  ( F `  z
) `'  <  ( F `  w )  <->  -.  ( F `  w
)  <  ( F `  z ) )
8382anbi2i 694 . . . . . . . . . . . . 13  |-  ( ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) )  <->  ( -.  ( F `  z )  <  ( F `  w
)  /\  -.  ( F `  w )  <  ( F `  z
) ) )
8478, 83bitr4i 252 . . . . . . . . . . . 12  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) )
8577, 84syl6ibr 227 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  ->  -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) ) ) )
8671, 85mt2d 117 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  -.  (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) ) )
8786ex 434 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
8855, 87sylbird 235 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
w  =/=  z  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
8988necon4ad 2677 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) )  ->  w  =  z ) )
9051, 89syl5bi 217 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( <. ( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>.  ->  w  =  z ) )
9148, 90sylbid 215 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  ->  w  =  z )
)
9291, 26syl6ib 226 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) )
9319, 28, 32, 33, 92wlogle 10107 . . 3  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
9493ralrimivva 2878 . 2  |-  ( ph  ->  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
95 dff13 6167 . 2  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  <->  ( T : ( 1 ... N ) --> ( NN 
X.  NN )  /\  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) ) )
9614, 94, 95sylanbrc 664 1  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811    C_ wss 3471   ~Pcpw 4015   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007    |` cres 5010   "cima 5011   -->wf 5590   -1-1->wf1 5591   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   supcsup 7918   RRcr 9508   1c1 9510    < clt 9645    <_ cle 9646   NNcn 10556   ...cfz 11697   #chash 12407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12408
This theorem is referenced by:  erdszelem10  28819
  Copyright terms: Public domain W3C validator