MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpre-sup Structured version   Unicode version

Theorem axpre-sup 9046
Description: A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 9153. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9070. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-sup  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
Distinct variable group:    x, y, z, A

Proof of Theorem axpre-sup
Dummy variables  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal2 9009 . . . . . . 7  |-  ( x  e.  RR  <->  ( ( 1st `  x )  e. 
R.  /\  x  =  <. ( 1st `  x
) ,  0R >. ) )
21simplbi 448 . . . . . 6  |-  ( x  e.  RR  ->  ( 1st `  x )  e. 
R. )
32adantl 454 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( 1st `  x
)  e.  R. )
4 fo1st 6368 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
5 fof 5655 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
6 ffn 5593 . . . . . . . . . . . 12  |-  ( 1st
: _V --> _V  ->  1st 
Fn  _V )
74, 5, 6mp2b 10 . . . . . . . . . . 11  |-  1st  Fn  _V
8 ssv 3370 . . . . . . . . . . 11  |-  A  C_  _V
9 fvelimab 5784 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( w  e.  ( 1st " A )  <->  E. y  e.  A  ( 1st `  y )  =  w ) )
107, 8, 9mp2an 655 . . . . . . . . . 10  |-  ( w  e.  ( 1st " A
)  <->  E. y  e.  A  ( 1st `  y )  =  w )
11 r19.29 2848 . . . . . . . . . . . 12  |-  ( ( A. y  e.  A  y  <RR  x  /\  E. y  e.  A  ( 1st `  y )  =  w )  ->  E. y  e.  A  ( y  <RR  x  /\  ( 1st `  y )  =  w ) )
12 ssel2 3345 . . . . . . . . . . . . . . . . 17  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  y  e.  RR )
13 ltresr2 9018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  RR  /\  x  e.  RR )  ->  ( y  <RR  x  <->  ( 1st `  y )  <R  ( 1st `  x ) ) )
14 breq1 4217 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  y )  =  w  ->  (
( 1st `  y
)  <R  ( 1st `  x
)  <->  w  <R  ( 1st `  x ) ) )
1513, 14sylan9bb 682 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  <-> 
w  <R  ( 1st `  x
) ) )
1615biimpd 200 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  ->  w  <R  ( 1st `  x ) ) )
1716exp31 589 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1812, 17syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1918imp4b 575 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( ( 1st `  y )  =  w  /\  y  <RR  x )  ->  w  <R  ( 1st `  x
) ) )
2019ancomsd 442 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( y 
<RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x ) ) )
2120an32s 781 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR  /\  x  e.  RR )  /\  y  e.  A
)  ->  ( (
y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2221rexlimdva 2832 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( E. y  e.  A  ( y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2311, 22syl5 31 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  (
( A. y  e.  A  y  <RR  x  /\  E. y  e.  A  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2423exp3a 427 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  ( E. y  e.  A  ( 1st `  y )  =  w  ->  w  <R  ( 1st `  x
) ) ) )
2510, 24syl7bi 223 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  (
w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x
) ) ) )
2625impr 604 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  e.  RR  /\  A. y  e.  A  y 
<RR  x ) )  -> 
( w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x ) ) )
2726adantlr 697 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  ( x  e.  RR  /\ 
A. y  e.  A  y  <RR  x ) )  ->  ( w  e.  ( 1st " A
)  ->  w  <R  ( 1st `  x ) ) )
2827ralrimiv 2790 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  ( x  e.  RR  /\ 
A. y  e.  A  y  <RR  x ) )  ->  A. w  e.  ( 1st " A ) w  <R  ( 1st `  x ) )
2928expr 600 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  A. w  e.  ( 1st " A ) w 
<R  ( 1st `  x
) ) )
30 breq2 4218 . . . . . . 7  |-  ( v  =  ( 1st `  x
)  ->  ( w  <R  v  <->  w  <R  ( 1st `  x ) ) )
3130ralbidv 2727 . . . . . 6  |-  ( v  =  ( 1st `  x
)  ->  ( A. w  e.  ( 1st " A ) w  <R  v  <->  A. w  e.  ( 1st " A ) w 
<R  ( 1st `  x
) ) )
3231rspcev 3054 . . . . 5  |-  ( ( ( 1st `  x
)  e.  R.  /\  A. w  e.  ( 1st " A ) w  <R  ( 1st `  x ) )  ->  E. v  e.  R.  A. w  e.  ( 1st " A
) w  <R  v
)
333, 29, 32ee12an 1373 . . . 4  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  E. v  e.  R.  A. w  e.  ( 1st " A ) w  <R  v ) )
3433rexlimdva 2832 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( E. x  e.  RR  A. y  e.  A  y 
<RR  x  ->  E. v  e.  R.  A. w  e.  ( 1st " A
) w  <R  v
) )
35 n0 3639 . . . . . 6  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
36 fnfvima 5978 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  A  C_  _V  /\  y  e.  A )  ->  ( 1st `  y )  e.  ( 1st " A
) )
377, 8, 36mp3an12 1270 . . . . . . . 8  |-  ( y  e.  A  ->  ( 1st `  y )  e.  ( 1st " A
) )
38 ne0i 3636 . . . . . . . 8  |-  ( ( 1st `  y )  e.  ( 1st " A
)  ->  ( 1st " A )  =/=  (/) )
3937, 38syl 16 . . . . . . 7  |-  ( y  e.  A  ->  ( 1st " A )  =/=  (/) )
4039exlimiv 1645 . . . . . 6  |-  ( E. y  y  e.  A  ->  ( 1st " A
)  =/=  (/) )
4135, 40sylbi 189 . . . . 5  |-  ( A  =/=  (/)  ->  ( 1st " A )  =/=  (/) )
42 supsr 8989 . . . . . 6  |-  ( ( ( 1st " A
)  =/=  (/)  /\  E. v  e.  R.  A. w  e.  ( 1st " A
) w  <R  v
)  ->  E. v  e.  R.  ( A. w  e.  ( 1st " A
)  -.  v  <R  w  /\  A. w  e. 
R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )
) )
4342ex 425 . . . . 5  |-  ( ( 1st " A )  =/=  (/)  ->  ( E. v  e.  R.  A. w  e.  ( 1st " A
) w  <R  v  ->  E. v  e.  R.  ( A. w  e.  ( 1st " A )  -.  v  <R  w  /\  A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u ) ) ) )
4441, 43syl 16 . . . 4  |-  ( A  =/=  (/)  ->  ( E. v  e.  R.  A. w  e.  ( 1st " A
) w  <R  v  ->  E. v  e.  R.  ( A. w  e.  ( 1st " A )  -.  v  <R  w  /\  A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u ) ) ) )
4544adantl 454 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( E. v  e.  R.  A. w  e.  ( 1st " A ) w  <R  v  ->  E. v  e.  R.  ( A. w  e.  ( 1st " A )  -.  v  <R  w  /\  A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u ) ) ) )
46 breq2 4218 . . . . . . . . . . . 12  |-  ( w  =  ( 1st `  y
)  ->  ( v  <R  w  <->  v  <R  ( 1st `  y ) ) )
4746notbid 287 . . . . . . . . . . 11  |-  ( w  =  ( 1st `  y
)  ->  ( -.  v  <R  w  <->  -.  v  <R  ( 1st `  y
) ) )
4847rspccv 3051 . . . . . . . . . 10  |-  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  ( ( 1st `  y )  e.  ( 1st " A
)  ->  -.  v  <R  ( 1st `  y
) ) )
4937, 48syl5com 29 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
5049adantl 454 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
51 elreal2 9009 . . . . . . . . . . . . 13  |-  ( y  e.  RR  <->  ( ( 1st `  y )  e. 
R.  /\  y  =  <. ( 1st `  y
) ,  0R >. ) )
5251simprbi 452 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  =  <. ( 1st `  y
) ,  0R >. )
5352breq2d 4226 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<-> 
<. v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >. ) )
54 ltresr 9017 . . . . . . . . . . 11  |-  ( <.
v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >.  <->  v  <R  ( 1st `  y
) )
5553, 54syl6bb 254 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5612, 55syl 16 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5756notbid 287 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( -.  <. v ,  0R >. 
<RR  y  <->  -.  v  <R  ( 1st `  y ) ) )
5850, 57sylibrd 227 . . . . . . 7  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -. 
<. v ,  0R >.  <RR  y ) )
5958ralrimdva 2798 . . . . . 6  |-  ( A 
C_  RR  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  A. y  e.  A  -.  <. v ,  0R >.  <RR  y ) )
6059ad2antrr 708 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  ->  ( A. w  e.  ( 1st " A
)  -.  v  <R  w  ->  A. y  e.  A  -.  <. v ,  0R >. 
<RR  y ) )
6152breq1d 4224 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <->  <. ( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >. ) )
62 ltresr 9017 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >.  <->  ( 1st `  y )  <R 
v )
6361, 62syl6bb 254 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <-> 
( 1st `  y
)  <R  v ) )
6451simplbi 448 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR  ->  ( 1st `  y )  e. 
R. )
65 breq1 4217 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  v  <->  ( 1st `  y
)  <R  v ) )
66 breq1 4217 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  u  <->  ( 1st `  y
)  <R  u ) )
6766rexbidv 2728 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( E. u  e.  ( 1st " A ) w  <R  u  <->  E. u  e.  ( 1st " A ) ( 1st `  y ) 
<R  u ) )
6865, 67imbi12d 313 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( 1st `  y
)  ->  ( (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  <->  ( ( 1st `  y )  <R  v  ->  E. u  e.  ( 1st " A ) ( 1st `  y
)  <R  u ) ) )
6968rspccv 3051 . . . . . . . . . . . . . . 15  |-  ( A. w  e.  R.  (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  ->  ( ( 1st `  y )  e. 
R.  ->  ( ( 1st `  y )  <R  v  ->  E. u  e.  ( 1st " A ) ( 1st `  y
)  <R  u ) ) )
7064, 69syl5 31 . . . . . . . . . . . . . 14  |-  ( A. w  e.  R.  (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  ->  ( y  e.  RR  ->  ( ( 1st `  y )  <R 
v  ->  E. u  e.  ( 1st " A
) ( 1st `  y
)  <R  u ) ) )
7170com3l 78 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  (
( 1st `  y
)  <R  v  ->  ( A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u )  ->  E. u  e.  ( 1st " A
) ( 1st `  y
)  <R  u ) ) )
7263, 71sylbid 208 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  ->  ( A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )  ->  E. u  e.  ( 1st " A ) ( 1st `  y
)  <R  u ) ) )
7372adantr 453 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( y  <RR  <. v ,  0R >.  ->  ( A. w  e.  R.  (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  ->  E. u  e.  ( 1st " A
) ( 1st `  y
)  <R  u ) ) )
74 fvelimab 5784 . . . . . . . . . . . . . . . 16  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( u  e.  ( 1st " A )  <->  E. z  e.  A  ( 1st `  z )  =  u ) )
757, 8, 74mp2an 655 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( 1st " A
)  <->  E. z  e.  A  ( 1st `  z )  =  u )
76 ssel2 3345 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
77 ltresr2 9018 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  <RR  z  <->  ( 1st `  y )  <R  ( 1st `  z ) ) )
7876, 77sylan2 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  ( 1st `  z
) ) )
79 breq2 4218 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  z )  =  u  ->  (
( 1st `  y
)  <R  ( 1st `  z
)  <->  ( 1st `  y
)  <R  u ) )
8078, 79sylan9bb 682 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A ) )  /\  ( 1st `  z )  =  u )  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  u ) )
8180exbiri 607 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( ( 1st `  z )  =  u  ->  ( ( 1st `  y )  <R  u  ->  y  <RR  z ) ) )
8281expr 600 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
( ( 1st `  y
)  <R  u  ->  y  <RR  z ) ) ) )
8382com4r 83 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  y ) 
<R  u  ->  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
y  <RR  z ) ) ) )
8483imp 420 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( z  e.  A  ->  ( ( 1st `  z )  =  u  ->  y  <RR  z ) ) )
8584reximdvai 2818 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( E. z  e.  A  ( 1st `  z )  =  u  ->  E. z  e.  A  y  <RR  z ) )
8675, 85syl5bi 210 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( u  e.  ( 1st " A
)  ->  E. z  e.  A  y  <RR  z ) )
8786expcom 426 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( ( 1st `  y
)  <R  u  ->  (
u  e.  ( 1st " A )  ->  E. z  e.  A  y  <RR  z ) ) )
8887com23 75 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( u  e.  ( 1st " A )  ->  ( ( 1st `  y )  <R  u  ->  E. z  e.  A  y  <RR  z ) ) )
8988rexlimdv 2831 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( E. u  e.  ( 1st " A
) ( 1st `  y
)  <R  u  ->  E. z  e.  A  y  <RR  z ) )
9073, 89syl6d 67 . . . . . . . . . 10  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( y  <RR  <. v ,  0R >.  ->  ( A. w  e.  R.  (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  ->  E. z  e.  A  y  <RR  z ) ) )
9190com23 75 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( A. w  e. 
R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )  ->  ( y  <RR  <. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) )
9291ex 425 . . . . . . . 8  |-  ( y  e.  RR  ->  ( A  C_  RR  ->  ( A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u )  ->  (
y  <RR  <. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) ) )
9392com3l 78 . . . . . . 7  |-  ( A 
C_  RR  ->  ( A. w  e.  R.  (
w  <R  v  ->  E. u  e.  ( 1st " A
) w  <R  u
)  ->  ( y  e.  RR  ->  ( y  <RR 
<. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) ) )
9493ad2antrr 708 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  ->  ( A. w  e. 
R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )  ->  ( y  e.  RR  ->  ( y  <RR  <. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) ) )
9594ralrimdv 2797 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  ->  ( A. w  e. 
R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )  ->  A. y  e.  RR  ( y  <RR  <. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) )
96 opelreal 9007 . . . . . . . 8  |-  ( <.
v ,  0R >.  e.  RR  <->  v  e.  R. )
9796biimpri 199 . . . . . . 7  |-  ( v  e.  R.  ->  <. v ,  0R >.  e.  RR )
9897adantl 454 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  -> 
<. v ,  0R >.  e.  RR )
99 breq1 4217 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( x  <RR  y  <->  <. v ,  0R >.  <RR  y ) )
10099notbid 287 . . . . . . . . . 10  |-  ( x  =  <. v ,  0R >.  ->  ( -.  x  <RR  y  <->  -.  <. v ,  0R >.  <RR  y ) )
101100ralbidv 2727 . . . . . . . . 9  |-  ( x  =  <. v ,  0R >.  ->  ( A. y  e.  A  -.  x  <RR  y  <->  A. y  e.  A  -.  <. v ,  0R >. 
<RR  y ) )
102 breq2 4218 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( y  <RR  x  <-> 
y  <RR  <. v ,  0R >. ) )
103102imbi1d 310 . . . . . . . . . 10  |-  ( x  =  <. v ,  0R >.  ->  ( ( y 
<RR  x  ->  E. z  e.  A  y  <RR  z )  <->  ( y  <RR  <.
v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) )
104103ralbidv 2727 . . . . . . . . 9  |-  ( x  =  <. v ,  0R >.  ->  ( A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z )  <->  A. y  e.  RR  ( y  <RR  <.
v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) )
105101, 104anbi12d 693 . . . . . . . 8  |-  ( x  =  <. v ,  0R >.  ->  ( ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) )  <->  ( A. y  e.  A  -.  <.
v ,  0R >.  <RR  y  /\  A. y  e.  RR  ( y  <RR  <.
v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) ) )
106105rspcev 3054 . . . . . . 7  |-  ( (
<. v ,  0R >.  e.  RR  /\  ( A. y  e.  A  -.  <.
v ,  0R >.  <RR  y  /\  A. y  e.  RR  ( y  <RR  <.
v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
107106ex 425 . . . . . 6  |-  ( <.
v ,  0R >.  e.  RR  ->  ( ( A. y  e.  A  -.  <. v ,  0R >. 
<RR  y  /\  A. y  e.  RR  ( y  <RR  <.
v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) ) )
10898, 107syl 16 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  ->  ( ( A. y  e.  A  -.  <. v ,  0R >.  <RR  y  /\  A. y  e.  RR  (
y  <RR  <. v ,  0R >.  ->  E. z  e.  A  y  <RR  z ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) ) )
10960, 95, 108syl2and 471 . . . 4  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  ->  ( ( A. w  e.  ( 1st " A
)  -.  v  <R  w  /\  A. w  e. 
R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w  <R  u )
)  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) ) )
110109rexlimdva 2832 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( E. v  e.  R.  ( A. w  e.  ( 1st " A )  -.  v  <R  w  /\  A. w  e.  R.  ( w  <R  v  ->  E. u  e.  ( 1st " A ) w 
<R  u ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) ) )
11134, 45, 1103syld 54 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( E. x  e.  RR  A. y  e.  A  y 
<RR  x  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) ) )
1121113impia 1151 1  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   (/)c0 3630   <.cop 3819   class class class wbr 4214   "cima 4883    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456   1stc1st 6349   R.cnr 8744   0Rc0r 8745    <R cltr 8750   RRcr 8991    <RR cltrr 8996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-omul 6731  df-er 6907  df-ec 6909  df-qs 6913  df-ni 8751  df-pli 8752  df-mi 8753  df-lti 8754  df-plpq 8787  df-mpq 8788  df-ltpq 8789  df-enq 8790  df-nq 8791  df-erq 8792  df-plq 8793  df-mq 8794  df-1nq 8795  df-rq 8796  df-ltnq 8797  df-np 8860  df-1p 8861  df-plp 8862  df-mp 8863  df-ltp 8864  df-plpr 8934  df-mpr 8935  df-enr 8936  df-nr 8937  df-plr 8938  df-mr 8939  df-ltr 8940  df-0r 8941  df-1r 8942  df-m1r 8943  df-r 9002  df-lt 9005
  Copyright terms: Public domain W3C validator