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Theorem erdszelem9 24664
Description: Lemma for erdsze 24667. (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 9089 . . . . . 6  |-  <  Or  RR
51, 2, 3, 4erdszelem6 24661 . . . . 5  |-  ( ph  ->  I : ( 1 ... N ) --> NN )
65ffvelrnda 5809 . . . 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 5351 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
94, 8mpbi 200 . . . . . 6  |-  `'  <  Or  RR
101, 2, 7, 9erdszelem6 24661 . . . . 5  |-  ( ph  ->  J : ( 1 ... N ) --> NN )
1110ffvelrnda 5809 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( J `  n )  e.  NN )
12 opelxpi 4850 . . . 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 5832 . 2  |-  ( ph  ->  T : ( 1 ... N ) --> ( NN  X.  NN ) )
16 fveq2 5668 . . . . . 6  |-  ( a  =  z  ->  ( T `  a )  =  ( T `  z ) )
17 fveq2 5668 . . . . . 6  |-  ( b  =  w  ->  ( T `  b )  =  ( T `  w ) )
1816, 17eqeqan12d 2402 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
19 eqeq12 2399 . . . . 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 5668 . . . . . . 7  |-  ( a  =  w  ->  ( T `  a )  =  ( T `  w ) )
22 fveq2 5668 . . . . . . 7  |-  ( b  =  z  ->  ( T `  b )  =  ( T `  z ) )
2321, 22eqeqan12d 2402 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  w
)  =  ( T `
 z ) ) )
24 eqcom 2389 . . . . . 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 2399 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
w  =  z ) )
27 eqcom 2389 . . . . . 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 10991 . . . . . . 7  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3130zred 10307 . . . . . 6  |-  ( z  e.  ( 1 ... N )  ->  z  e.  RR )
3231ssriv 3295 . . . . 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 5668 . . . . . . . . . 10  |-  ( n  =  z  ->  (
I `  n )  =  ( I `  z ) )
37 fveq2 5668 . . . . . . . . . 10  |-  ( n  =  z  ->  ( J `  n )  =  ( J `  z ) )
3836, 37opeq12d 3934 . . . . . . . . 9  |-  ( n  =  z  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  z ) ,  ( J `  z ) >. )
39 opex 4368 . . . . . . . . 9  |-  <. (
I `  z ) ,  ( J `  z ) >.  e.  _V
4038, 14, 39fvmpt 5745 . . . . . . . 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 5668 . . . . . . . . . 10  |-  ( n  =  w  ->  (
I `  n )  =  ( I `  w ) )
44 fveq2 5668 . . . . . . . . . 10  |-  ( n  =  w  ->  ( J `  n )  =  ( J `  w ) )
4543, 44opeq12d 3934 . . . . . . . . 9  |-  ( n  =  w  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  w ) ,  ( J `  w ) >. )
46 opex 4368 . . . . . . . . 9  |-  <. (
I `  w ) ,  ( J `  w ) >.  e.  _V
4745, 14, 46fvmpt 5745 . . . . . . . 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 2401 . . . . . 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 5682 . . . . . . . 8  |-  ( I `
 z )  e. 
_V
51 fvex 5682 . . . . . . . 8  |-  ( J `
 z )  e. 
_V
5250, 51opth 4376 . . . . . . 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 3289 . . . . . . . . . 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 9156 . . . . . . . . 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 5947 . . . . . . . . . . . . . . . . 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 2585 . . . . . . . . . . . . . 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 5579 . . . . . . . . . . . . . . . 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 5810 . . . . . . . . . . . . 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 5810 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  w )  e.  RR )
7168, 70lttri2d 9144 . . . . . . . . . . . 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 24663 . . . . . . . . . . . . 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 24663 . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . . 16  |-  ( F `
 z )  e. 
_V
81 fvex 5682 . . . . . . . . . . . . . . . 16  |-  ( F `
 w )  e. 
_V
8280, 81brcnv 4995 . . . . . . . . . . . . . . 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 2611 . . . . . . 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 9492 . . 3  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
9594ralrimivva 2741 . 2  |-  ( ph  ->  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
96 dff13 5943 . 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 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653    C_ wss 3263   ~Pcpw 3742   <.cop 3760   class class class wbr 4153    e. cmpt 4207    Or wor 4443    X. cxp 4816   `'ccnv 4817    |` cres 4820   "cima 4821   -->wf 5390   -1-1->wf1 5391   ` cfv 5394    Isom wiso 5395  (class class class)co 6020   supcsup 7380   RRcr 8922   1c1 8924    < clt 9053    <_ cle 9054   NNcn 9932   ...cfz 10975   #chash 11545
This theorem is referenced by:  erdszelem10  24665
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-hash 11546
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