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Theorem axpre-sup 9332
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 9446. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9356. (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 9295 . . . . . . 7  |-  ( x  e.  RR  <->  ( ( 1st `  x )  e. 
R.  /\  x  =  <. ( 1st `  x
) ,  0R >. ) )
21simplbi 457 . . . . . 6  |-  ( x  e.  RR  ->  ( 1st `  x )  e. 
R. )
32adantl 463 . . . . 5  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  x  e.  RR )  ->  ( 1st `  x
)  e.  R. )
4 fo1st 6595 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
5 fof 5617 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
6 ffn 5556 . . . . . . . . . . . 12  |-  ( 1st
: _V --> _V  ->  1st 
Fn  _V )
74, 5, 6mp2b 10 . . . . . . . . . . 11  |-  1st  Fn  _V
8 ssv 3373 . . . . . . . . . . 11  |-  A  C_  _V
9 fvelimab 5744 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( w  e.  ( 1st " A )  <->  E. y  e.  A  ( 1st `  y )  =  w ) )
107, 8, 9mp2an 667 . . . . . . . . . 10  |-  ( w  e.  ( 1st " A
)  <->  E. y  e.  A  ( 1st `  y )  =  w )
11 r19.29 2855 . . . . . . . . . . . 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 3348 . . . . . . . . . . . . . . . . 17  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  y  e.  RR )
13 ltresr2 9304 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  RR  /\  x  e.  RR )  ->  ( y  <RR  x  <->  ( 1st `  y )  <R  ( 1st `  x ) ) )
14 breq1 4292 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  y )  =  w  ->  (
( 1st `  y
)  <R  ( 1st `  x
)  <->  w  <R  ( 1st `  x ) ) )
1513, 14sylan9bb 694 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  <-> 
w  <R  ( 1st `  x
) ) )
1615biimpd 207 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  RR  /\  x  e.  RR )  /\  ( 1st `  y
)  =  w )  ->  ( y  <RR  x  ->  w  <R  ( 1st `  x ) ) )
1716exp31 601 . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( ( 1st `  y )  =  w  /\  y  <RR  x )  ->  w  <R  ( 1st `  x
) ) )
2019ancomsd 451 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  RR  /\  y  e.  A )  /\  x  e.  RR )  ->  ( ( y 
<RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x ) ) )
2120an32s 797 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR  /\  x  e.  RR )  /\  y  e.  A
)  ->  ( (
y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2221rexlimdva 2839 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( E. y  e.  A  ( y  <RR  x  /\  ( 1st `  y )  =  w )  ->  w  <R  ( 1st `  x
) ) )
2311, 22syl5 32 . . . . . . . . . . 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 436 . . . . . . . . . 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 230 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  ( A. y  e.  A  y  <RR  x  ->  (
w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x
) ) ) )
2625impr 616 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  e.  RR  /\  A. y  e.  A  y 
<RR  x ) )  -> 
( w  e.  ( 1st " A )  ->  w  <R  ( 1st `  x ) ) )
2726adantlr 709 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  ( x  e.  RR  /\ 
A. y  e.  A  y  <RR  x ) )  ->  ( w  e.  ( 1st " A
)  ->  w  <R  ( 1st `  x ) ) )
2827ralrimiv 2796 . . . . . 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 612 . . . . 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 4293 . . . . . . 7  |-  ( v  =  ( 1st `  x
)  ->  ( w  <R  v  <->  w  <R  ( 1st `  x ) ) )
3130ralbidv 2733 . . . . . 6  |-  ( v  =  ( 1st `  x
)  ->  ( A. w  e.  ( 1st " A ) w  <R  v  <->  A. w  e.  ( 1st " A ) w 
<R  ( 1st `  x
) ) )
3231rspcev 3070 . . . . 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 542 . . . 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 2839 . . 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 3643 . . . . . 6  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
36 fnfvima 5952 . . . . . . . . 9  |-  ( ( 1st  Fn  _V  /\  A  C_  _V  /\  y  e.  A )  ->  ( 1st `  y )  e.  ( 1st " A
) )
377, 8, 36mp3an12 1299 . . . . . . . 8  |-  ( y  e.  A  ->  ( 1st `  y )  e.  ( 1st " A
) )
38 ne0i 3640 . . . . . . . 8  |-  ( ( 1st `  y )  e.  ( 1st " A
)  ->  ( 1st " A )  =/=  (/) )
3937, 38syl 16 . . . . . . 7  |-  ( y  e.  A  ->  ( 1st " A )  =/=  (/) )
4039exlimiv 1693 . . . . . 6  |-  ( E. y  y  e.  A  ->  ( 1st " A
)  =/=  (/) )
4135, 40sylbi 195 . . . . 5  |-  ( A  =/=  (/)  ->  ( 1st " A )  =/=  (/) )
42 supsr 9275 . . . . . 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 434 . . . . 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 463 . . 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 4293 . . . . . . . . . . . 12  |-  ( w  =  ( 1st `  y
)  ->  ( v  <R  w  <->  v  <R  ( 1st `  y ) ) )
4746notbid 294 . . . . . . . . . . 11  |-  ( w  =  ( 1st `  y
)  ->  ( -.  v  <R  w  <->  -.  v  <R  ( 1st `  y
) ) )
4847rspccv 3067 . . . . . . . . . 10  |-  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  ( ( 1st `  y )  e.  ( 1st " A
)  ->  -.  v  <R  ( 1st `  y
) ) )
4937, 48syl5com 30 . . . . . . . . 9  |-  ( y  e.  A  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
5049adantl 463 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -.  v  <R  ( 1st `  y ) ) )
51 elreal2 9295 . . . . . . . . . . . . 13  |-  ( y  e.  RR  <->  ( ( 1st `  y )  e. 
R.  /\  y  =  <. ( 1st `  y
) ,  0R >. ) )
5251simprbi 461 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  =  <. ( 1st `  y
) ,  0R >. )
5352breq2d 4301 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  ( <. v ,  0R >.  <RR  y 
<-> 
<. v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >. ) )
54 ltresr 9303 . . . . . . . . . . 11  |-  ( <.
v ,  0R >.  <RR  <. ( 1st `  y
) ,  0R >.  <->  v  <R  ( 1st `  y
) )
5553, 54syl6bb 261 . . . . . . . . . 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 294 . . . . . . . 8  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( -.  <. v ,  0R >. 
<RR  y  <->  -.  v  <R  ( 1st `  y ) ) )
5850, 57sylibrd 234 . . . . . . 7  |-  ( ( A  C_  RR  /\  y  e.  A )  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  -. 
<. v ,  0R >.  <RR  y ) )
5958ralrimdva 2804 . . . . . 6  |-  ( A 
C_  RR  ->  ( A. w  e.  ( 1st " A )  -.  v  <R  w  ->  A. y  e.  A  -.  <. v ,  0R >.  <RR  y ) )
6059ad2antrr 720 . . . . 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 4299 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <->  <. ( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >. ) )
62 ltresr 9303 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  y
) ,  0R >.  <RR  <. v ,  0R >.  <->  ( 1st `  y )  <R 
v )
6361, 62syl6bb 261 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  (
y  <RR  <. v ,  0R >.  <-> 
( 1st `  y
)  <R  v ) )
6451simplbi 457 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR  ->  ( 1st `  y )  e. 
R. )
65 breq1 4292 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  v  <->  ( 1st `  y
)  <R  v ) )
66 breq1 4292 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( 1st `  y
)  ->  ( w  <R  u  <->  ( 1st `  y
)  <R  u ) )
6766rexbidv 2734 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( 1st `  y
)  ->  ( E. u  e.  ( 1st " A ) w  <R  u  <->  E. u  e.  ( 1st " A ) ( 1st `  y ) 
<R  u ) )
6865, 67imbi12d 320 . . . . . . . . . . . . . . . 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 3067 . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 5744 . . . . . . . . . . . . . . . 16  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( u  e.  ( 1st " A )  <->  E. z  e.  A  ( 1st `  z )  =  u ) )
757, 8, 74mp2an 667 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( 1st " A
)  <->  E. z  e.  A  ( 1st `  z )  =  u )
76 ssel2 3348 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
77 ltresr2 9304 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  <RR  z  <->  ( 1st `  y )  <R  ( 1st `  z ) ) )
7876, 77sylan2 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  ( 1st `  z
) ) )
79 breq2 4293 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  z )  =  u  ->  (
( 1st `  y
)  <R  ( 1st `  z
)  <->  ( 1st `  y
)  <R  u ) )
8078, 79sylan9bb 694 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A ) )  /\  ( 1st `  z )  =  u )  ->  ( y  <RR  z  <->  ( 1st `  y
)  <R  u ) )
8180exbiri 619 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  /\  ( A  C_  RR  /\  z  e.  A )
)  ->  ( ( 1st `  z )  =  u  ->  ( ( 1st `  y )  <R  u  ->  y  <RR  z ) ) )
8281expr 612 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
( ( 1st `  y
)  <R  u  ->  y  <RR  z ) ) ) )
8382com4r 86 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  y ) 
<R  u  ->  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( z  e.  A  ->  ( ( 1st `  z
)  =  u  -> 
y  <RR  z ) ) ) )
8483imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( z  e.  A  ->  ( ( 1st `  z )  =  u  ->  y  <RR  z ) ) )
8584reximdvai 2824 . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  y
)  <R  u  /\  (
y  e.  RR  /\  A  C_  RR ) )  ->  ( u  e.  ( 1st " A
)  ->  E. z  e.  A  y  <RR  z ) )
8786expcom 435 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( ( 1st `  y
)  <R  u  ->  (
u  e.  ( 1st " A )  ->  E. z  e.  A  y  <RR  z ) ) )
8887com23 78 . . . . . . . . . . . 12  |-  ( ( y  e.  RR  /\  A  C_  RR )  -> 
( u  e.  ( 1st " A )  ->  ( ( 1st `  y )  <R  u  ->  E. z  e.  A  y  <RR  z ) ) )
8988rexlimdv 2838 . . . . . . . . . . 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 69 . . . . . . . . . 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 78 . . . . . . . . 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 434 . . . . . . . 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 81 . . . . . . 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 720 . . . . . 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 2803 . . . . 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 9293 . . . . . . . 8  |-  ( <.
v ,  0R >.  e.  RR  <->  v  e.  R. )
9796biimpri 206 . . . . . . 7  |-  ( v  e.  R.  ->  <. v ,  0R >.  e.  RR )
9897adantl 463 . . . . . 6  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  v  e.  R. )  -> 
<. v ,  0R >.  e.  RR )
99 breq1 4292 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( x  <RR  y  <->  <. v ,  0R >.  <RR  y ) )
10099notbid 294 . . . . . . . . . 10  |-  ( x  =  <. v ,  0R >.  ->  ( -.  x  <RR  y  <->  -.  <. v ,  0R >.  <RR  y ) )
101100ralbidv 2733 . . . . . . . . 9  |-  ( x  =  <. v ,  0R >.  ->  ( A. y  e.  A  -.  x  <RR  y  <->  A. y  e.  A  -.  <. v ,  0R >. 
<RR  y ) )
102 breq2 4293 . . . . . . . . . . 11  |-  ( x  =  <. v ,  0R >.  ->  ( y  <RR  x  <-> 
y  <RR  <. v ,  0R >. ) )
103102imbi1d 317 . . . . . . . . . 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 2733 . . . . . . . . 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 705 . . . . . . . 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 3070 . . . . . . 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 434 . . . . . 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 480 . . . 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 2839 . . 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 55 . 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 1179 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 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325   (/)c0 3634   <.cop 3880   class class class wbr 4289   "cima 4839    Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415   1stc1st 6574   R.cnr 9030   0Rc0r 9031    <R cltr 9036   RRcr 9277    <RR cltrr 9282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-ltpq 9075  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081  df-rq 9082  df-ltnq 9083  df-np 9146  df-1p 9147  df-plp 9148  df-mp 9149  df-ltp 9150  df-plpr 9220  df-mpr 9221  df-enr 9222  df-nr 9223  df-plr 9224  df-mr 9225  df-ltr 9226  df-0r 9227  df-1r 9228  df-m1r 9229  df-r 9288  df-lt 9291
This theorem is referenced by: (None)
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