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Theorem nocvxmin 25651
Description: Given a non-empty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
Assertion
Ref Expression
nocvxmin  |-  ( ( A  =/=  (/)  /\  A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x < s z  /\  z < s y )  ->  z  e.  A
) )  ->  E! w  e.  A  ( bday `  w )  = 
|^| ( bday " A
) )
Distinct variable group:    w, A, x, y, z

Proof of Theorem nocvxmin
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 imassrn 5219 . . . . . 6  |-  ( bday " A )  C_  ran  bday
2 bdayrn 25637 . . . . . 6  |-  ran  bday  =  On
31, 2sseqtri 3382 . . . . 5  |-  ( bday " A )  C_  On
4 bdaydm 25638 . . . . . . . . . . 11  |-  dom  bday  =  No
54sseq2i 3375 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  <->  A  C_  No )
6 bdayfun 25636 . . . . . . . . . . 11  |-  Fun  bday
7 funfvima2 5977 . . . . . . . . . . 11  |-  ( ( Fun  bday  /\  A  C_  dom  bday )  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
86, 7mpan 653 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
95, 8sylbir 206 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( x  e.  A  ->  ( bday `  x )  e.  ( bday " A
) ) )
10 elex2 2970 . . . . . . . . 9  |-  ( (
bday `  x )  e.  ( bday " A
)  ->  E. w  w  e.  ( bday " A ) )
119, 10syl6 32 . . . . . . . 8  |-  ( A 
C_  No  ->  ( x  e.  A  ->  E. w  w  e.  ( bday " A ) ) )
1211exlimdv 1647 . . . . . . 7  |-  ( A 
C_  No  ->  ( E. x  x  e.  A  ->  E. w  w  e.  ( bday " A
) ) )
13 n0 3639 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 n0 3639 . . . . . . 7  |-  ( (
bday " A )  =/=  (/) 
<->  E. w  w  e.  ( bday " A
) )
1512, 13, 143imtr4g 263 . . . . . 6  |-  ( A 
C_  No  ->  ( A  =/=  (/)  ->  ( bday " A )  =/=  (/) ) )
1615impcom 421 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( bday " A )  =/=  (/) )
17 onint 4778 . . . . 5  |-  ( ( ( bday " A
)  C_  On  /\  ( bday " A )  =/=  (/) )  ->  |^| ( bday " A )  e.  ( bday " A
) )
183, 16, 17sylancr 646 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  |^| ( bday " A )  e.  ( bday " A
) )
19 bdayfn 25639 . . . . . 6  |-  bday  Fn  No
20 fvelimab 5785 . . . . . 6  |-  ( (
bday  Fn  No  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2119, 20mpan 653 . . . . 5  |-  ( A 
C_  No  ->  ( |^| ( bday " A )  e.  ( bday " A
)  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A
) ) )
2221adantl 454 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2318, 22mpbid 203 . . 3  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
24233adant3 978 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x < s z  /\  z < s y )  ->  z  e.  A
) )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
25 ssel 3344 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( w  e.  A  ->  w  e.  No ) )
26 ssel 3344 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( t  e.  A  ->  t  e.  No ) )
2725, 26anim12d 548 . . . . . . . 8  |-  ( A 
C_  No  ->  ( ( w  e.  A  /\  t  e.  A )  ->  ( w  e.  No  /\  t  e.  No ) ) )
2827imp 420 . . . . . . 7  |-  ( ( A  C_  No  /\  (
w  e.  A  /\  t  e.  A )
)  ->  ( w  e.  No  /\  t  e.  No ) )
2928ad2ant2r 729 . . . . . 6  |-  ( ( ( A  C_  No  /\ 
A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x <
s z  /\  z
< s y )  -> 
z  e.  A ) )  /\  ( ( w  e.  A  /\  t  e.  A )  /\  ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) ) ) )  ->  ( w  e.  No  /\  t  e.  No ) )
30 nocvxminlem 25650 . . . . . . 7  |-  ( ( A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  (
( x < s
z  /\  z < s y )  ->  z  e.  A ) )  -> 
( ( ( w  e.  A  /\  t  e.  A )  /\  (
( bday `  w )  =  |^| ( bday " A
)  /\  ( bday `  t )  =  |^| ( bday " A ) ) )  ->  -.  w < s t ) )
3130imp 420 . . . . . 6  |-  ( ( ( A  C_  No  /\ 
A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x <
s z  /\  z
< s y )  -> 
z  e.  A ) )  /\  ( ( w  e.  A  /\  t  e.  A )  /\  ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) ) ) )  ->  -.  w < s t )
32 ancom 439 . . . . . . . . 9  |-  ( ( w  e.  A  /\  t  e.  A )  <->  ( t  e.  A  /\  w  e.  A )
)
33 ancom 439 . . . . . . . . 9  |-  ( ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  <->  ( ( bday `  t )  = 
|^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )
3432, 33anbi12i 680 . . . . . . . 8  |-  ( ( ( w  e.  A  /\  t  e.  A
)  /\  ( ( bday `  w )  = 
|^| ( bday " A
)  /\  ( bday `  t )  =  |^| ( bday " A ) ) )  <->  ( (
t  e.  A  /\  w  e.  A )  /\  ( ( bday `  t
)  =  |^| ( bday " A )  /\  ( bday `  w )  =  |^| ( bday " A
) ) ) )
35 nocvxminlem 25650 . . . . . . . 8  |-  ( ( A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  (
( x < s
z  /\  z < s y )  ->  z  e.  A ) )  -> 
( ( ( t  e.  A  /\  w  e.  A )  /\  (
( bday `  t )  =  |^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )  ->  -.  t < s w ) )
3634, 35syl5bi 210 . . . . . . 7  |-  ( ( A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  (
( x < s
z  /\  z < s y )  ->  z  e.  A ) )  -> 
( ( ( w  e.  A  /\  t  e.  A )  /\  (
( bday `  w )  =  |^| ( bday " A
)  /\  ( bday `  t )  =  |^| ( bday " A ) ) )  ->  -.  t < s w ) )
3736imp 420 . . . . . 6  |-  ( ( ( A  C_  No  /\ 
A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x <
s z  /\  z
< s y )  -> 
z  e.  A ) )  /\  ( ( w  e.  A  /\  t  e.  A )  /\  ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) ) ) )  ->  -.  t < s w )
38 slttrieq2 25634 . . . . . . 7  |-  ( ( w  e.  No  /\  t  e.  No )  ->  ( w  =  t  <-> 
( -.  w <
s t  /\  -.  t < s w ) ) )
3938biimpar 473 . . . . . 6  |-  ( ( ( w  e.  No  /\  t  e.  No )  /\  ( -.  w < s t  /\  -.  t < s w ) )  ->  w  =  t )
4029, 31, 37, 39syl12anc 1183 . . . . 5  |-  ( ( ( A  C_  No  /\ 
A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x <
s z  /\  z
< s y )  -> 
z  e.  A ) )  /\  ( ( w  e.  A  /\  t  e.  A )  /\  ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) ) ) )  ->  w  =  t )
4140exp32 590 . . . 4  |-  ( ( A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  (
( x < s
z  /\  z < s y )  ->  z  e.  A ) )  -> 
( ( w  e.  A  /\  t  e.  A )  ->  (
( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  ->  w  =  t ) ) )
4241ralrimivv 2799 . . 3  |-  ( ( A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  (
( x < s
z  /\  z < s y )  ->  z  e.  A ) )  ->  A. w  e.  A  A. t  e.  A  ( ( ( bday `  w )  =  |^| ( bday " A )  /\  ( bday `  t
)  =  |^| ( bday " A ) )  ->  w  =  t ) )
43423adant1 976 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x < s z  /\  z < s y )  ->  z  e.  A
) )  ->  A. w  e.  A  A. t  e.  A  ( (
( bday `  w )  =  |^| ( bday " A
)  /\  ( bday `  t )  =  |^| ( bday " A ) )  ->  w  =  t ) )
44 fveq2 5731 . . . 4  |-  ( w  =  t  ->  ( bday `  w )  =  ( bday `  t
) )
4544eqeq1d 2446 . . 3  |-  ( w  =  t  ->  (
( bday `  w )  =  |^| ( bday " A
)  <->  ( bday `  t
)  =  |^| ( bday " A ) ) )
4645reu4 3130 . 2  |-  ( E! w  e.  A  (
bday `  w )  =  |^| ( bday " A
)  <->  ( E. w  e.  A  ( bday `  w )  =  |^| ( bday " A )  /\  A. w  e.  A  A. t  e.  A  ( ( (
bday `  w )  =  |^| ( bday " A
)  /\  ( bday `  t )  =  |^| ( bday " A ) )  ->  w  =  t ) ) )
4724, 43, 46sylanbrc 647 1  |-  ( ( A  =/=  (/)  /\  A  C_  No  /\  A. x  e.  A  A. y  e.  A  A. z  e.  No  ( ( x < s z  /\  z < s y )  ->  z  e.  A
) )  ->  E! w  e.  A  ( bday `  w )  = 
|^| ( bday " A
) )
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   E!wreu 2709    C_ wss 3322   (/)c0 3630   |^|cint 4052   class class class wbr 4215   Oncon0 4584   dom cdm 4881   ran crn 4882   "cima 4884   Fun wfun 5451    Fn wfn 5452   ` cfv 5457   Nocsur 25600   < scslt 25601   bdaycbday 25602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-2o 6728  df-no 25603  df-slt 25604  df-bday 25605
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