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Theorem nocvxmin 25563
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 5179 . . . . . 6  |-  ( bday " A )  C_  ran  bday
2 bdayrn 25549 . . . . . 6  |-  ran  bday  =  On
31, 2sseqtri 3344 . . . . 5  |-  ( bday " A )  C_  On
4 bdaydm 25550 . . . . . . . . . . 11  |-  dom  bday  =  No
54sseq2i 3337 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  <->  A  C_  No )
6 bdayfun 25548 . . . . . . . . . . 11  |-  Fun  bday
7 funfvima2 5937 . . . . . . . . . . 11  |-  ( ( Fun  bday  /\  A  C_  dom  bday )  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
86, 7mpan 652 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
95, 8sylbir 205 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( x  e.  A  ->  ( bday `  x )  e.  ( bday " A
) ) )
10 elex2 2932 . . . . . . . . 9  |-  ( (
bday `  x )  e.  ( bday " A
)  ->  E. w  w  e.  ( bday " A ) )
119, 10syl6 31 . . . . . . . 8  |-  ( A 
C_  No  ->  ( x  e.  A  ->  E. w  w  e.  ( bday " A ) ) )
1211exlimdv 1643 . . . . . . 7  |-  ( A 
C_  No  ->  ( E. x  x  e.  A  ->  E. w  w  e.  ( bday " A
) ) )
13 n0 3601 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 n0 3601 . . . . . . 7  |-  ( (
bday " A )  =/=  (/) 
<->  E. w  w  e.  ( bday " A
) )
1512, 13, 143imtr4g 262 . . . . . 6  |-  ( A 
C_  No  ->  ( A  =/=  (/)  ->  ( bday " A )  =/=  (/) ) )
1615impcom 420 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( bday " A )  =/=  (/) )
17 onint 4738 . . . . 5  |-  ( ( ( bday " A
)  C_  On  /\  ( bday " A )  =/=  (/) )  ->  |^| ( bday " A )  e.  ( bday " A
) )
183, 16, 17sylancr 645 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  |^| ( bday " A )  e.  ( bday " A
) )
19 bdayfn 25551 . . . . . 6  |-  bday  Fn  No
20 fvelimab 5745 . . . . . 6  |-  ( (
bday  Fn  No  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2119, 20mpan 652 . . . . 5  |-  ( A 
C_  No  ->  ( |^| ( bday " A )  e.  ( bday " A
)  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A
) ) )
2221adantl 453 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2318, 22mpbid 202 . . 3  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
24233adant3 977 . 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 3306 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( w  e.  A  ->  w  e.  No ) )
26 ssel 3306 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( t  e.  A  ->  t  e.  No ) )
2725, 26anim12d 547 . . . . . . . 8  |-  ( A 
C_  No  ->  ( ( w  e.  A  /\  t  e.  A )  ->  ( w  e.  No  /\  t  e.  No ) ) )
2827imp 419 . . . . . . 7  |-  ( ( A  C_  No  /\  (
w  e.  A  /\  t  e.  A )
)  ->  ( w  e.  No  /\  t  e.  No ) )
2928ad2ant2r 728 . . . . . 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 25562 . . . . . . 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 419 . . . . . 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 438 . . . . . . . . 9  |-  ( ( w  e.  A  /\  t  e.  A )  <->  ( t  e.  A  /\  w  e.  A )
)
33 ancom 438 . . . . . . . . 9  |-  ( ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  <->  ( ( bday `  t )  = 
|^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )
3432, 33anbi12i 679 . . . . . . . 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 25562 . . . . . . . 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 209 . . . . . . 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 419 . . . . . 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 25546 . . . . . . 7  |-  ( ( w  e.  No  /\  t  e.  No )  ->  ( w  =  t  <-> 
( -.  w <
s t  /\  -.  t < s w ) ) )
3938biimpar 472 . . . . . 6  |-  ( ( ( w  e.  No  /\  t  e.  No )  /\  ( -.  w < s t  /\  -.  t < s w ) )  ->  w  =  t )
4029, 31, 37, 39syl12anc 1182 . . . . 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 589 . . . 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 2761 . . 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 975 . 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 5691 . . . 4  |-  ( w  =  t  ->  ( bday `  w )  =  ( bday `  t
) )
4544eqeq1d 2416 . . 3  |-  ( w  =  t  ->  (
( bday `  w )  =  |^| ( bday " A
)  <->  ( bday `  t
)  =  |^| ( bday " A ) ) )
4645reu4 3092 . 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 646 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   E!wreu 2672    C_ wss 3284   (/)c0 3592   |^|cint 4014   class class class wbr 4176   Oncon0 4545   dom cdm 4841   ran crn 4842   "cima 4844   Fun wfun 5411    Fn wfn 5412   ` cfv 5417   Nocsur 25512   < scslt 25513   bdaycbday 25514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-1o 6687  df-2o 6688  df-no 25515  df-slt 25516  df-bday 25517
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