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Theorem nocvxmin 29426
Description: Given a nonempty 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 5338 . . . . . 6  |-  ( bday " A )  C_  ran  bday
2 bdayrn 29412 . . . . . 6  |-  ran  bday  =  On
31, 2sseqtri 3521 . . . . 5  |-  ( bday " A )  C_  On
4 bdaydm 29413 . . . . . . . . . . 11  |-  dom  bday  =  No
54sseq2i 3514 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  <->  A  C_  No )
6 bdayfun 29411 . . . . . . . . . . 11  |-  Fun  bday
7 funfvima2 6133 . . . . . . . . . . 11  |-  ( ( Fun  bday  /\  A  C_  dom  bday )  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
86, 7mpan 670 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
95, 8sylbir 213 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( x  e.  A  ->  ( bday `  x )  e.  ( bday " A
) ) )
10 elex2 3107 . . . . . . . . 9  |-  ( (
bday `  x )  e.  ( bday " A
)  ->  E. w  w  e.  ( bday " A ) )
119, 10syl6 33 . . . . . . . 8  |-  ( A 
C_  No  ->  ( x  e.  A  ->  E. w  w  e.  ( bday " A ) ) )
1211exlimdv 1711 . . . . . . 7  |-  ( A 
C_  No  ->  ( E. x  x  e.  A  ->  E. w  w  e.  ( bday " A
) ) )
13 n0 3780 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 n0 3780 . . . . . . 7  |-  ( (
bday " A )  =/=  (/) 
<->  E. w  w  e.  ( bday " A
) )
1512, 13, 143imtr4g 270 . . . . . 6  |-  ( A 
C_  No  ->  ( A  =/=  (/)  ->  ( bday " A )  =/=  (/) ) )
1615impcom 430 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( bday " A )  =/=  (/) )
17 onint 6615 . . . . 5  |-  ( ( ( bday " A
)  C_  On  /\  ( bday " A )  =/=  (/) )  ->  |^| ( bday " A )  e.  ( bday " A
) )
183, 16, 17sylancr 663 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  |^| ( bday " A )  e.  ( bday " A
) )
19 bdayfn 29414 . . . . . 6  |-  bday  Fn  No
20 fvelimab 5914 . . . . . 6  |-  ( (
bday  Fn  No  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2119, 20mpan 670 . . . . 5  |-  ( A 
C_  No  ->  ( |^| ( bday " A )  e.  ( bday " A
)  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A
) ) )
2221adantl 466 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2318, 22mpbid 210 . . 3  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
24233adant3 1017 . 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 3483 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( w  e.  A  ->  w  e.  No ) )
26 ssel 3483 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( t  e.  A  ->  t  e.  No ) )
2725, 26anim12d 563 . . . . . . . 8  |-  ( A 
C_  No  ->  ( ( w  e.  A  /\  t  e.  A )  ->  ( w  e.  No  /\  t  e.  No ) ) )
2827imp 429 . . . . . . 7  |-  ( ( A  C_  No  /\  (
w  e.  A  /\  t  e.  A )
)  ->  ( w  e.  No  /\  t  e.  No ) )
2928ad2ant2r 746 . . . . . 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 29425 . . . . . . 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 429 . . . . . 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 450 . . . . . . . . 9  |-  ( ( w  e.  A  /\  t  e.  A )  <->  ( t  e.  A  /\  w  e.  A )
)
33 ancom 450 . . . . . . . . 9  |-  ( ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  <->  ( ( bday `  t )  = 
|^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )
3432, 33anbi12i 697 . . . . . . . 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 29425 . . . . . . . 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 217 . . . . . . 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 429 . . . . . 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 29409 . . . . . . 7  |-  ( ( w  e.  No  /\  t  e.  No )  ->  ( w  =  t  <-> 
( -.  w <s t  /\  -.  t <s w ) ) )
3938biimpar 485 . . . . . 6  |-  ( ( ( w  e.  No  /\  t  e.  No )  /\  ( -.  w <s t  /\  -.  t <s w ) )  ->  w  =  t )
4029, 31, 37, 39syl12anc 1227 . . . . 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 605 . . . 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 2863 . . 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 1015 . 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 5856 . . . 4  |-  ( w  =  t  ->  ( bday `  w )  =  ( bday `  t
) )
4544eqeq1d 2445 . . 3  |-  ( w  =  t  ->  (
( bday `  w )  =  |^| ( bday " A
)  <->  ( bday `  t
)  =  |^| ( bday " A ) ) )
4645reu4 3279 . 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 664 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   E!wreu 2795    C_ wss 3461   (/)c0 3770   |^|cint 4271   class class class wbr 4437   Oncon0 4868   dom cdm 4989   ran crn 4990   "cima 4992   Fun wfun 5572    Fn wfn 5573   ` cfv 5578   Nocsur 29375   <scslt 29376   bdaycbday 29377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-1o 7132  df-2o 7133  df-no 29378  df-slt 29379  df-bday 29380
This theorem is referenced by: (None)
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