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Theorem axpre-sup 9601
Description: A nonempty, 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 9717. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9625. (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 9564 . . . . . . 7  |-  ( x  e.  RR  <->  ( ( 1st `  x )  e. 
R.  /\  x  =  <. ( 1st `  x
) ,  0R >. ) )
21simplbi 461 . . . . . 6  |-  ( x  e.  RR  ->  ( 1st `  x )  e. 
R. )
32adantl 467 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( 1st `  x
)  e.  R. )
4 fo1st 6828 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
5 fof 5810 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
6 ffn 5746 . . . . . . . . . . . 12  |-  ( 1st
: _V --> _V  ->  1st 
Fn  _V )
74, 5, 6mp2b 10 . . . . . . . . . . 11  |-  1st  Fn  _V
8 ssv 3484 . . . . . . . . . . 11  |-  A  C_  _V
9 fvelimab 5938 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( w  e.  ( 1st " A )  <->  E. y  e.  A  ( 1st `  y )  =  w ) )
107, 8, 9mp2an 676 . . . . . . . . . 10  |-  ( w  e.  ( 1st " A
)  <->  E. y  e.  A  ( 1st `  y )  =  w )
11 r19.29 2960 . . . . . . . . . . . 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 3459 . . . . . . . . . . . . . . . . 17  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  y  e.  RR )
13 ltresr2 9573 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  RR  /\  x  e.  RR )  ->  ( y  <RR  x  <->  ( 1st `  y )  <R  ( 1st `  x ) ) )
14 breq1 4426 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  y )  =  w  ->  (
( 1st `  y
)  <R  ( 1st `  x
)  <->  w  <R  ( 1st `  x ) ) )
1513, 14sylan9bb 704 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  <-> 
w  <R  ( 1st `  x
) ) )
1615biimpd 210 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  ->  w  <R  ( 1st `  x ) ) )
1716exp31 607 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1812, 17syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  (
x  e.  RR  ->  ( ( 1st `  y
)  =  w  -> 
( y  <RR  x  ->  w  <R  ( 1st `  x
) ) ) ) )
1918imp4b 593 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( ( 1st `  y )  =  w  /\  y  <RR  x )  ->  w  <R  ( 1st `  x
) ) )
2019ancomsd 455 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( y 
<RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x ) ) )
2120an32s 811 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR  /\  x  e.  RR )  /\  y  e.  A
)  ->  ( (
y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2221rexlimdva 2914 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( E. y  e.  A  ( y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2311, 22syl5 33 . . . . . . . . . . 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
) ) )
2423expd 437 . . . . . . . . . 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 233 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  (
w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x
) ) ) )
2625impr 623 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  e.  RR  /\  A. y  e.  A  y 
<RR  x ) )  -> 
( w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x ) ) )
2726adantlr 719 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  ( x  e.  RR  /\ 
A. y  e.  A  y  <RR  x ) )  ->  ( w  e.  ( 1st " A
)  ->  w  <R  ( 1st `  x ) ) )
2827ralrimiv 2834 . . . . . 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 618 . . . . 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 4427 . . . . . . 7  |-  ( v  =  ( 1st `  x
)  ->  ( w  <R  v  <->  w  <R  ( 1st `  x ) ) )
3130ralbidv 2861 . . . . . 6  |-  ( v  =  ( 1st `  x
)  ->  ( A. w  e.  ( 1st " A ) w  <R  v  <->  A. w  e.  ( 1st " A ) w 
<R  ( 1st `  x
) ) )
3231rspcev 3182 . . . . 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, 32syl6an 547 . . . 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 2914 . . 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 3771 . . . . . 6  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
36 fnfvima 6159 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  A  C_  _V  /\  y  e.  A )  ->  ( 1st `  y )  e.  ( 1st " A
) )
377, 8, 36mp3an12 1350 . . . . . . . 8  |-  ( y  e.  A  ->  ( 1st `  y )  e.  ( 1st " A
) )
38 ne0i 3767 . . . . . . . 8  |-  ( ( 1st `  y )  e.  ( 1st " A
)  ->  ( 1st " A )  =/=  (/) )
3937, 38syl 17 . . . . . . 7  |-  ( y  e.  A  ->  ( 1st " A )  =/=  (/) )
4039exlimiv 1770 . . . . . 6  |-  ( E. y  y  e.  A  ->  ( 1st " A
)  =/=  (/) )
4135, 40sylbi 198 . . . . 5  |-  ( A  =/=  (/)  ->  ( 1st " A )  =/=  (/) )
42 supsr 9544 . . . . . 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 435 . . . . 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 17 . . . 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 467 . . 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 4427 . . . . . . . . . . . 12  |-  ( w  =  ( 1st `  y
)  ->  ( v  <R  w  <->  v  <R  ( 1st `  y ) ) )
4746notbid 295 . . . . . . . . . . 11  |-  ( w  =  ( 1st `  y
)  ->  ( -.  v  <R  w  <->  -.  v  <R  ( 1st `  y
) ) )
4847rspccv 3179 . . . . . . . . . 10  |-  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  ( ( 1st `  y )  e.  ( 1st " A
)  ->  -.  v  <R  ( 1st `  y
) ) )
4937, 48syl5com 31 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
5049adantl 467 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
51 elreal2 9564 . . . . . . . . . . . . 13  |-  ( y  e.  RR  <->  ( ( 1st `  y )  e. 
R.  /\  y  =  <. ( 1st `  y
) ,  0R >. ) )
5251simprbi 465 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  =  <. ( 1st `  y
) ,  0R >. )
5352breq2d 4435 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<-> 
<. v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >. ) )
54 ltresr 9572 . . . . . . . . . . 11  |-  ( <.
v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >.  <->  v  <R  ( 1st `  y
) )
5553, 54syl6bb 264 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5612, 55syl 17 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( <. v ,  0R >.  <RR  y 
<->  v  <R  ( 1st `  y ) ) )
5756notbid 295 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( -.  <. v ,  0R >. 
<RR  y  <->  -.  v  <R  ( 1st `  y ) ) )
5850, 57sylibrd 237 . . . . . . 7  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -. 
<. v ,  0R >.  <RR  y ) )
5958ralrimdva 2840 . . . . . 6  |-  ( A 
C_  RR  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  A. y  e.  A  -.  <. v ,  0R >.  <RR  y ) )
6059ad2antrr 730 . . . . 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 4433 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <->  <. ( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >. ) )
62 ltresr 9572 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >.  <->  ( 1st `  y )  <R 
v )
6361, 62syl6bb 264 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <-> 
( 1st `  y
)  <R  v ) )
6451simplbi 461 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR  ->  ( 1st `  y )  e. 
R. )
65 breq1 4426 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  v  <->  ( 1st `  y
)  <R  v ) )
66 breq1 4426 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  u  <->  ( 1st `  y
)  <R  u ) )
6766rexbidv 2936 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( E. u  e.  ( 1st " A ) w  <R  u  <->  E. u  e.  ( 1st " A ) ( 1st `  y ) 
<R  u ) )
6865, 67imbi12d 321 . . . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . 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 218 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 5938 . . . . . . . . . . . . . . . 16  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( u  e.  ( 1st " A )  <->  E. z  e.  A  ( 1st `  z )  =  u ) )
757, 8, 74mp2an 676 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( 1st " A
)  <->  E. z  e.  A  ( 1st `  z )  =  u )
76 ssel2 3459 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
77 ltresr2 9573 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  <RR  z  <->  ( 1st `  y )  <R  ( 1st `  z ) ) )
7876, 77sylan2 476 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  ( 1st `  z
) ) )
79 breq2 4427 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  z )  =  u  ->  (
( 1st `  y
)  <R  ( 1st `  z
)  <->  ( 1st `  y
)  <R  u ) )
8078, 79sylan9bb 704 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A ) )  /\  ( 1st `  z )  =  u )  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  u ) )
8180exbiri 626 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( ( 1st `  z )  =  u  ->  ( ( 1st `  y )  <R  u  ->  y  <RR  z ) ) )
8281expr 618 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
( ( 1st `  y
)  <R  u  ->  y  <RR  z ) ) ) )
8382com4r 89 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  y ) 
<R  u  ->  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
y  <RR  z ) ) ) )
8483imp 430 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( z  e.  A  ->  ( ( 1st `  z )  =  u  ->  y  <RR  z ) ) )
8584reximdvai 2894 . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( u  e.  ( 1st " A
)  ->  E. z  e.  A  y  <RR  z ) )
8786expcom 436 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( ( 1st `  y
)  <R  u  ->  (
u  e.  ( 1st " A )  ->  E. z  e.  A  y  <RR  z ) ) )
8887com23 81 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( u  e.  ( 1st " A )  ->  ( ( 1st `  y )  <R  u  ->  E. z  e.  A  y  <RR  z ) ) )
8988rexlimdv 2912 . . . . . . . . . . 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 71 . . . . . . . . . 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 81 . . . . . . . . 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 435 . . . . . . . 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 84 . . . . . . 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 730 . . . . . 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 2838 . . . . 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 9562 . . . . . . . 8  |-  ( <.
v ,  0R >.  e.  RR  <->  v  e.  R. )
9796biimpri 209 . . . . . . 7  |-  ( v  e.  R.  ->  <. v ,  0R >.  e.  RR )
9897adantl 467 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  -> 
<. v ,  0R >.  e.  RR )
99 breq1 4426 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( x  <RR  y  <->  <. v ,  0R >.  <RR  y ) )
10099notbid 295 . . . . . . . . . 10  |-  ( x  =  <. v ,  0R >.  ->  ( -.  x  <RR  y  <->  -.  <. v ,  0R >.  <RR  y ) )
101100ralbidv 2861 . . . . . . . . 9  |-  ( x  =  <. v ,  0R >.  ->  ( A. y  e.  A  -.  x  <RR  y  <->  A. y  e.  A  -.  <. v ,  0R >. 
<RR  y ) )
102 breq2 4427 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( y  <RR  x  <-> 
y  <RR  <. v ,  0R >. ) )
103102imbi1d 318 . . . . . . . . . 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 2861 . . . . . . . . 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 715 . . . . . . . 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 3182 . . . . . . 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 435 . . . . . 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 17 . . . . 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 485 . . . 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 2914 . . 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 57 . 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 1202 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    C_ wss 3436   (/)c0 3761   <.cop 4004   class class class wbr 4423   "cima 4856    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601   1stc1st 6806   R.cnr 9298   0Rc0r 9299    <R cltr 9304   RRcr 9546    <RR cltrr 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-omul 7199  df-er 7375  df-ec 7377  df-qs 7381  df-ni 9305  df-pli 9306  df-mi 9307  df-lti 9308  df-plpq 9341  df-mpq 9342  df-ltpq 9343  df-enq 9344  df-nq 9345  df-erq 9346  df-plq 9347  df-mq 9348  df-1nq 9349  df-rq 9350  df-ltnq 9351  df-np 9414  df-1p 9415  df-plp 9416  df-mp 9417  df-ltp 9418  df-enr 9488  df-nr 9489  df-plr 9490  df-mr 9491  df-ltr 9492  df-0r 9493  df-1r 9494  df-m1r 9495  df-r 9557  df-lt 9560
This theorem is referenced by: (None)
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