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Theorem axpre-sup 8807
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 8914. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8831. (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 8770 . . . . . . 7  |-  ( x  e.  RR  <->  ( ( 1st `  x )  e. 
R.  /\  x  =  <. ( 1st `  x
) ,  0R >. ) )
21simplbi 446 . . . . . 6  |-  ( x  e.  RR  ->  ( 1st `  x )  e. 
R. )
32adantl 452 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( 1st `  x
)  e.  R. )
4 fo1st 6155 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
5 fof 5467 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
6 ffn 5405 . . . . . . . . . . . 12  |-  ( 1st
: _V --> _V  ->  1st 
Fn  _V )
74, 5, 6mp2b 9 . . . . . . . . . . 11  |-  1st  Fn  _V
8 ssv 3211 . . . . . . . . . . 11  |-  A  C_  _V
9 fvelimab 5594 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( w  e.  ( 1st " A )  <->  E. y  e.  A  ( 1st `  y )  =  w ) )
107, 8, 9mp2an 653 . . . . . . . . . 10  |-  ( w  e.  ( 1st " A
)  <->  E. y  e.  A  ( 1st `  y )  =  w )
11 r19.29 2696 . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . . . 17  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  y  e.  RR )
13 ltresr2 8779 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  RR  /\  x  e.  RR )  ->  ( y  <RR  x  <->  ( 1st `  y )  <R  ( 1st `  x ) ) )
14 breq1 4042 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  y )  =  w  ->  (
( 1st `  y
)  <R  ( 1st `  x
)  <->  w  <R  ( 1st `  x ) ) )
1513, 14sylan9bb 680 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  <-> 
w  <R  ( 1st `  x
) ) )
1615biimpd 198 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  ->  w  <R  ( 1st `  x ) ) )
1716exp31 587 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1812, 17syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1918imp4b 573 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( ( 1st `  y )  =  w  /\  y  <RR  x )  ->  w  <R  ( 1st `  x
) ) )
2019ancomsd 440 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( y 
<RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x ) ) )
2120an32s 779 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR  /\  x  e.  RR )  /\  y  e.  A
)  ->  ( (
y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2221rexlimdva 2680 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( E. y  e.  A  ( y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2311, 22syl5 28 . . . . . . . . . . 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 425 . . . . . . . . . 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 221 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  (
w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x
) ) ) )
2625impr 602 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  e.  RR  /\  A. y  e.  A  y 
<RR  x ) )  -> 
( w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x ) ) )
2726adantlr 695 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  ( x  e.  RR  /\ 
A. y  e.  A  y  <RR  x ) )  ->  ( w  e.  ( 1st " A
)  ->  w  <R  ( 1st `  x ) ) )
2827ralrimiv 2638 . . . . . 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 598 . . . . 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 4043 . . . . . . 7  |-  ( v  =  ( 1st `  x
)  ->  ( w  <R  v  <->  w  <R  ( 1st `  x ) ) )
3130ralbidv 2576 . . . . . 6  |-  ( v  =  ( 1st `  x
)  ->  ( A. w  e.  ( 1st " A ) w  <R  v  <->  A. w  e.  ( 1st " A ) w 
<R  ( 1st `  x
) ) )
3231rspcev 2897 . . . . 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 1353 . . . 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 2680 . . 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 3477 . . . . . 6  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
36 fnfvima 5772 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  A  C_  _V  /\  y  e.  A )  ->  ( 1st `  y )  e.  ( 1st " A
) )
377, 8, 36mp3an12 1267 . . . . . . . 8  |-  ( y  e.  A  ->  ( 1st `  y )  e.  ( 1st " A
) )
38 ne0i 3474 . . . . . . . 8  |-  ( ( 1st `  y )  e.  ( 1st " A
)  ->  ( 1st " A )  =/=  (/) )
3937, 38syl 15 . . . . . . 7  |-  ( y  e.  A  ->  ( 1st " A )  =/=  (/) )
4039exlimiv 1624 . . . . . 6  |-  ( E. y  y  e.  A  ->  ( 1st " A
)  =/=  (/) )
4135, 40sylbi 187 . . . . 5  |-  ( A  =/=  (/)  ->  ( 1st " A )  =/=  (/) )
42 supsr 8750 . . . . . 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 423 . . . . 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 15 . . . 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 452 . . 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 4043 . . . . . . . . . . . 12  |-  ( w  =  ( 1st `  y
)  ->  ( v  <R  w  <->  v  <R  ( 1st `  y ) ) )
4746notbid 285 . . . . . . . . . . 11  |-  ( w  =  ( 1st `  y
)  ->  ( -.  v  <R  w  <->  -.  v  <R  ( 1st `  y
) ) )
4847rspccv 2894 . . . . . . . . . 10  |-  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  ( ( 1st `  y )  e.  ( 1st " A
)  ->  -.  v  <R  ( 1st `  y
) ) )
4937, 48syl5com 26 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
5049adantl 452 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
51 elreal2 8770 . . . . . . . . . . . . 13  |-  ( y  e.  RR  <->  ( ( 1st `  y )  e. 
R.  /\  y  =  <. ( 1st `  y
) ,  0R >. ) )
5251simprbi 450 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  =  <. ( 1st `  y
) ,  0R >. )
5352breq2d 4051 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<-> 
<. v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >. ) )
54 ltresr 8778 . . . . . . . . . . 11  |-  ( <.
v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >.  <->  v  <R  ( 1st `  y
) )
5553, 54syl6bb 252 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5612, 55syl 15 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5756notbid 285 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( -.  <. v ,  0R >. 
<RR  y  <->  -.  v  <R  ( 1st `  y ) ) )
5850, 57sylibrd 225 . . . . . . 7  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -. 
<. v ,  0R >.  <RR  y ) )
5958ralrimdva 2646 . . . . . 6  |-  ( A 
C_  RR  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  A. y  e.  A  -.  <. v ,  0R >.  <RR  y ) )
6059ad2antrr 706 . . . . 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 4049 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <->  <. ( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >. ) )
62 ltresr 8778 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >.  <->  ( 1st `  y )  <R 
v )
6361, 62syl6bb 252 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <-> 
( 1st `  y
)  <R  v ) )
6451simplbi 446 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR  ->  ( 1st `  y )  e. 
R. )
65 breq1 4042 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  v  <->  ( 1st `  y
)  <R  v ) )
66 breq1 4042 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  u  <->  ( 1st `  y
)  <R  u ) )
6766rexbidv 2577 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( E. u  e.  ( 1st " A ) w  <R  u  <->  E. u  e.  ( 1st " A ) ( 1st `  y ) 
<R  u ) )
6865, 67imbi12d 311 . . . . . . . . . . . . . . . 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 2894 . . . . . . . . . . . . . . 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 28 . . . . . . . . . . . . . 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 75 . . . . . . . . . . . . 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 206 . . . . . . . . . . . 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 451 . . . . . . . . . . 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 5594 . . . . . . . . . . . . . . . 16  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( u  e.  ( 1st " A )  <->  E. z  e.  A  ( 1st `  z )  =  u ) )
757, 8, 74mp2an 653 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( 1st " A
)  <->  E. z  e.  A  ( 1st `  z )  =  u )
76 ssel2 3188 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
77 ltresr2 8779 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  <RR  z  <->  ( 1st `  y )  <R  ( 1st `  z ) ) )
7876, 77sylan2 460 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  ( 1st `  z
) ) )
79 breq2 4043 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  z )  =  u  ->  (
( 1st `  y
)  <R  ( 1st `  z
)  <->  ( 1st `  y
)  <R  u ) )
8078, 79sylan9bb 680 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A ) )  /\  ( 1st `  z )  =  u )  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  u ) )
8180exbiri 605 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( ( 1st `  z )  =  u  ->  ( ( 1st `  y )  <R  u  ->  y  <RR  z ) ) )
8281expr 598 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
( ( 1st `  y
)  <R  u  ->  y  <RR  z ) ) ) )
8382com4r 80 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  y ) 
<R  u  ->  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
y  <RR  z ) ) ) )
8483imp 418 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( z  e.  A  ->  ( ( 1st `  z )  =  u  ->  y  <RR  z ) ) )
8584reximdvai 2666 . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( u  e.  ( 1st " A
)  ->  E. z  e.  A  y  <RR  z ) )
8786expcom 424 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( ( 1st `  y
)  <R  u  ->  (
u  e.  ( 1st " A )  ->  E. z  e.  A  y  <RR  z ) ) )
8887com23 72 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( u  e.  ( 1st " A )  ->  ( ( 1st `  y )  <R  u  ->  E. z  e.  A  y  <RR  z ) ) )
8988rexlimdv 2679 . . . . . . . . . . 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 64 . . . . . . . . . 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 72 . . . . . . . . 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 423 . . . . . . . 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 75 . . . . . . 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 706 . . . . . 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 2645 . . . . 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 8768 . . . . . . . 8  |-  ( <.
v ,  0R >.  e.  RR  <->  v  e.  R. )
9796biimpri 197 . . . . . . 7  |-  ( v  e.  R.  ->  <. v ,  0R >.  e.  RR )
9897adantl 452 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  -> 
<. v ,  0R >.  e.  RR )
99 breq1 4042 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( x  <RR  y  <->  <. v ,  0R >.  <RR  y ) )
10099notbid 285 . . . . . . . . . 10  |-  ( x  =  <. v ,  0R >.  ->  ( -.  x  <RR  y  <->  -.  <. v ,  0R >.  <RR  y ) )
101100ralbidv 2576 . . . . . . . . 9  |-  ( x  =  <. v ,  0R >.  ->  ( A. y  e.  A  -.  x  <RR  y  <->  A. y  e.  A  -.  <. v ,  0R >. 
<RR  y ) )
102 breq2 4043 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( y  <RR  x  <-> 
y  <RR  <. v ,  0R >. ) )
103102imbi1d 308 . . . . . . . . . 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 2576 . . . . . . . . 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 691 . . . . . . . 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 2897 . . . . . . 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 423 . . . . . 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 15 . . . . 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 469 . . . 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 2680 . . 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 51 . 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 1148 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 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039   "cima 4708    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   R.cnr 8505   0Rc0r 8506    <R cltr 8511   RRcr 8752    <RR cltrr 8757
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-inf2 7358
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-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-plp 8623  df-mp 8624  df-ltp 8625  df-plpr 8695  df-mpr 8696  df-enr 8697  df-nr 8698  df-plr 8699  df-mr 8700  df-ltr 8701  df-0r 8702  df-1r 8703  df-m1r 8704  df-r 8763  df-lt 8766
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