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Theorem erdszelem9 24877
Description: Lemma for erdsze 24880. (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 9148 . . . . . 6  |-  <  Or  RR
51, 2, 3, 4erdszelem6 24874 . . . . 5  |-  ( ph  ->  I : ( 1 ... N ) --> NN )
65ffvelrnda 5862 . . . 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 cnvso 5403 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
94, 8mpbi 200 . . . . . 6  |-  `'  <  Or  RR
101, 2, 7, 9erdszelem6 24874 . . . . 5  |-  ( ph  ->  J : ( 1 ... N ) --> NN )
1110ffvelrnda 5862 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( J `  n )  e.  NN )
12 opelxpi 4902 . . . 4  |-  ( ( ( I `  n
)  e.  NN  /\  ( J `  n )  e.  NN )  ->  <. ( I `  n
) ,  ( J `
 n ) >.  e.  ( NN  X.  NN ) )
136, 11, 12syl2anc 643 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  <. (
I `  n ) ,  ( J `  n ) >.  e.  ( NN  X.  NN ) )
14 erdszelem.t . . 3  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
1513, 14fmptd 5885 . 2  |-  ( ph  ->  T : ( 1 ... N ) --> ( NN  X.  NN ) )
16 fveq2 5720 . . . . . 6  |-  ( a  =  z  ->  ( T `  a )  =  ( T `  z ) )
17 fveq2 5720 . . . . . 6  |-  ( b  =  w  ->  ( T `  b )  =  ( T `  w ) )
1816, 17eqeqan12d 2450 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
19 eqeq12 2447 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( a  =  b  <-> 
z  =  w ) )
2018, 19imbi12d 312 . . . 4  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
21 fveq2 5720 . . . . . . 7  |-  ( a  =  w  ->  ( T `  a )  =  ( T `  w ) )
22 fveq2 5720 . . . . . . 7  |-  ( b  =  z  ->  ( T `  b )  =  ( T `  z ) )
2321, 22eqeqan12d 2450 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  w
)  =  ( T `
 z ) ) )
24 eqcom 2437 . . . . . 6  |-  ( ( T `  w )  =  ( T `  z )  <->  ( T `  z )  =  ( T `  w ) )
2523, 24syl6bb 253 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
26 eqeq12 2447 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
w  =  z ) )
27 eqcom 2437 . . . . . 6  |-  ( w  =  z  <->  z  =  w )
2826, 27syl6bb 253 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
z  =  w ) )
2925, 28imbi12d 312 . . . 4  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
30 elfzelz 11051 . . . . . . 7  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3130zred 10367 . . . . . 6  |-  ( z  e.  ( 1 ... N )  ->  z  e.  RR )
3231ssriv 3344 . . . . 5  |-  ( 1 ... N )  C_  RR
3332a1i 11 . . . 4  |-  ( ph  ->  ( 1 ... N
)  C_  RR )
34 biidd 229 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( ( T `
 z )  =  ( T `  w
)  ->  z  =  w )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
35 simpr1 963 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  ( 1 ... N
) )
36 fveq2 5720 . . . . . . . . . 10  |-  ( n  =  z  ->  (
I `  n )  =  ( I `  z ) )
37 fveq2 5720 . . . . . . . . . 10  |-  ( n  =  z  ->  ( J `  n )  =  ( J `  z ) )
3836, 37opeq12d 3984 . . . . . . . . 9  |-  ( n  =  z  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  z ) ,  ( J `  z ) >. )
39 opex 4419 . . . . . . . . 9  |-  <. (
I `  z ) ,  ( J `  z ) >.  e.  _V
4038, 14, 39fvmpt 5798 . . . . . . . 8  |-  ( z  e.  ( 1 ... N )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
4135, 40syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
42 simpr2 964 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  ( 1 ... N
) )
43 fveq2 5720 . . . . . . . . . 10  |-  ( n  =  w  ->  (
I `  n )  =  ( I `  w ) )
44 fveq2 5720 . . . . . . . . . 10  |-  ( n  =  w  ->  ( J `  n )  =  ( J `  w ) )
4543, 44opeq12d 3984 . . . . . . . . 9  |-  ( n  =  w  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  w ) ,  ( J `  w ) >. )
46 opex 4419 . . . . . . . . 9  |-  <. (
I `  w ) ,  ( J `  w ) >.  e.  _V
4745, 14, 46fvmpt 5798 . . . . . . . 8  |-  ( w  e.  ( 1 ... N )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4842, 47syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4941, 48eqeq12d 2449 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  <->  <. ( I `
 z ) ,  ( J `  z
) >.  =  <. (
I `  w ) ,  ( J `  w ) >. )
)
50 fvex 5734 . . . . . . . 8  |-  ( I `
 z )  e. 
_V
51 fvex 5734 . . . . . . . 8  |-  ( J `
 z )  e. 
_V
5250, 51opth 4427 . . . . . . 7  |-  ( <.
( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>. 
<->  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) )
5335, 31syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  RR )
5432, 42sseldi 3338 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  RR )
55 simpr3 965 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  <_  w )
5653, 54, 55leltned 9216 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  w  =/=  z ) )
572adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  F : ( 1 ... N ) -1-1-> RR )
58 f1fveq 6000 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( 1 ... N ) -1-1-> RR  /\  ( z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N ) ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
5957, 35, 42, 58syl12anc 1182 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
6059, 27syl6bbr 255 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  w  =  z ) )
6160necon3bid 2633 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =/=  ( F `
 w )  <->  w  =/=  z ) )
6256, 61bitr4d 248 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  ( F `  z )  =/=  ( F `  w )
) )
6362biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  =/=  ( F `  w )
)
64 f1f 5631 . . . . . . . . . . . . . . . 16  |-  ( F : ( 1 ... N ) -1-1-> RR  ->  F : ( 1 ... N ) --> RR )
652, 64syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 1 ... N ) --> RR )
6665ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) --> RR )
6735adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  e.  ( 1 ... N
) )
6866, 67ffvelrnd 5863 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  e.  RR )
6942adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  w  e.  ( 1 ... N
) )
7066, 69ffvelrnd 5863 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  w )  e.  RR )
7168, 70lttri2d 9204 . . . . . . . . . . . 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
) ) ) )
7263, 71mpbid 202 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  <  ( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) )
731ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  N  e.  NN )
742ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) -1-1-> RR )
75 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  <  w )
7673, 74, 3, 4, 67, 69, 75erdszelem8 24876 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
I `  z )  =  ( I `  w )  ->  -.  ( F `  z )  <  ( F `  w ) ) )
7773, 74, 7, 9, 67, 69, 75erdszelem8 24876 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( J `  z )  =  ( J `  w )  ->  -.  ( F `  z ) `'  <  ( F `  w ) ) )
7876, 77anim12d 547 . . . . . . . . . . . 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 ) ) ) )
79 ioran 477 . . . . . . . . . . . . 13  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  w )  <  ( F `  z ) ) )
80 fvex 5734 . . . . . . . . . . . . . . . 16  |-  ( F `
 z )  e. 
_V
81 fvex 5734 . . . . . . . . . . . . . . . 16  |-  ( F `
 w )  e. 
_V
8280, 81brcnv 5047 . . . . . . . . . . . . . . 15  |-  ( ( F `  z ) `'  <  ( F `  w )  <->  ( F `  w )  <  ( F `  z )
)
8382notbii 288 . . . . . . . . . . . . . 14  |-  ( -.  ( F `  z
) `'  <  ( F `  w )  <->  -.  ( F `  w
)  <  ( F `  z ) )
8483anbi2i 676 . . . . . . . . . . . . 13  |-  ( ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) )  <->  ( -.  ( F `  z )  <  ( F `  w
)  /\  -.  ( F `  w )  <  ( F `  z
) ) )
8579, 84bitr4i 244 . . . . . . . . . . . 12  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) )
8678, 85syl6ibr 219 . . . . . . . . . . 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 ) ) ) )
8772, 86mt2d 111 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  -.  (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) ) )
8887ex 424 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
8956, 88sylbird 227 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
w  =/=  z  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
9089necon4ad 2659 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) )  ->  w  =  z ) )
9152, 90syl5bi 209 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( <. ( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>.  ->  w  =  z ) )
9249, 91sylbid 207 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  ->  w  =  z )
)
9392, 27syl6ib 218 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) )
9420, 29, 33, 34, 93wlogle 9552 . . 3  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
9594ralrimivva 2790 . 2  |-  ( ph  ->  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
96 dff13 5996 . 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 ) ) )
9715, 95, 96sylanbrc 646 1  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701    C_ wss 3312   ~Pcpw 3791   <.cop 3809   class class class wbr 4204    e. cmpt 4258    Or wor 4494    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873   -->wf 5442   -1-1->wf1 5443   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   supcsup 7437   RRcr 8981   1c1 8983    < clt 9112    <_ cle 9113   NNcn 9992   ...cfz 11035   #chash 11610
This theorem is referenced by:  erdszelem10  24878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611
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