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Theorem nocvxmin 29691
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 5336 . . . . . 6  |-  ( bday " A )  C_  ran  bday
2 bdayrn 29677 . . . . . 6  |-  ran  bday  =  On
31, 2sseqtri 3521 . . . . 5  |-  ( bday " A )  C_  On
4 bdaydm 29678 . . . . . . . . . . 11  |-  dom  bday  =  No
54sseq2i 3514 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  <->  A  C_  No )
6 bdayfun 29676 . . . . . . . . . . 11  |-  Fun  bday
7 funfvima2 6123 . . . . . . . . . . 11  |-  ( ( Fun  bday  /\  A  C_  dom  bday )  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
86, 7mpan 668 . . . . . . . . . 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 3118 . . . . . . . . 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 1729 . . . . . . 7  |-  ( A 
C_  No  ->  ( E. x  x  e.  A  ->  E. w  w  e.  ( bday " A
) ) )
13 n0 3793 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 n0 3793 . . . . . . 7  |-  ( (
bday " A )  =/=  (/) 
<->  E. w  w  e.  ( bday " A
) )
1512, 13, 143imtr4g 270 . . . . . 6  |-  ( A 
C_  No  ->  ( A  =/=  (/)  ->  ( bday " A )  =/=  (/) ) )
1615impcom 428 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( bday " A )  =/=  (/) )
17 onint 6603 . . . . 5  |-  ( ( ( bday " A
)  C_  On  /\  ( bday " A )  =/=  (/) )  ->  |^| ( bday " A )  e.  ( bday " A
) )
183, 16, 17sylancr 661 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  |^| ( bday " A )  e.  ( bday " A
) )
19 bdayfn 29679 . . . . . 6  |-  bday  Fn  No
20 fvelimab 5904 . . . . . 6  |-  ( (
bday  Fn  No  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2119, 20mpan 668 . . . . 5  |-  ( A 
C_  No  ->  ( |^| ( bday " A )  e.  ( bday " A
)  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A
) ) )
2221adantl 464 . . . 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 1014 . 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 561 . . . . . . . 8  |-  ( A 
C_  No  ->  ( ( w  e.  A  /\  t  e.  A )  ->  ( w  e.  No  /\  t  e.  No ) ) )
2827imp 427 . . . . . . 7  |-  ( ( A  C_  No  /\  (
w  e.  A  /\  t  e.  A )
)  ->  ( w  e.  No  /\  t  e.  No ) )
2928ad2ant2r 744 . . . . . 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 29690 . . . . . . 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 427 . . . . . 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 448 . . . . . . . . 9  |-  ( ( w  e.  A  /\  t  e.  A )  <->  ( t  e.  A  /\  w  e.  A )
)
33 ancom 448 . . . . . . . . 9  |-  ( ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  <->  ( ( bday `  t )  = 
|^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )
3432, 33anbi12i 695 . . . . . . . 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 29690 . . . . . . . 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 427 . . . . . 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 29674 . . . . . . 7  |-  ( ( w  e.  No  /\  t  e.  No )  ->  ( w  =  t  <-> 
( -.  w <s t  /\  -.  t <s w ) ) )
3938biimpar 483 . . . . . 6  |-  ( ( ( w  e.  No  /\  t  e.  No )  /\  ( -.  w <s t  /\  -.  t <s w ) )  ->  w  =  t )
4029, 31, 37, 39syl12anc 1224 . . . . 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 603 . . . 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 2874 . . 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 1012 . 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 5848 . . . 4  |-  ( w  =  t  ->  ( bday `  w )  =  ( bday `  t
) )
4544eqeq1d 2456 . . 3  |-  ( w  =  t  ->  (
( bday `  w )  =  |^| ( bday " A
)  <->  ( bday `  t
)  =  |^| ( bday " A ) ) )
4645reu4 3290 . 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 662 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 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   E!wreu 2806    C_ wss 3461   (/)c0 3783   |^|cint 4271   class class class wbr 4439   Oncon0 4867   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564    Fn wfn 5565   ` cfv 5570   Nocsur 29640   <scslt 29641   bdaycbday 29642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-2o 7123  df-no 29643  df-slt 29644  df-bday 29645
This theorem is referenced by: (None)
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