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Theorem nocvxmin 24903
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 5104 . . . . . 6  |-  ( bday " A )  C_  ran  bday
2 bdayrn 24889 . . . . . 6  |-  ran  bday  =  On
31, 2sseqtri 3286 . . . . 5  |-  ( bday " A )  C_  On
4 bdaydm 24890 . . . . . . . . . . 11  |-  dom  bday  =  No
54sseq2i 3279 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  <->  A  C_  No )
6 bdayfun 24888 . . . . . . . . . . 11  |-  Fun  bday
7 funfvima2 5837 . . . . . . . . . . 11  |-  ( ( Fun  bday  /\  A  C_  dom  bday )  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
86, 7mpan 651 . . . . . . . . . 10  |-  ( A 
C_  dom  bday  ->  (
x  e.  A  -> 
( bday `  x )  e.  ( bday " A
) ) )
95, 8sylbir 204 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( x  e.  A  ->  ( bday `  x )  e.  ( bday " A
) ) )
10 elex2 2876 . . . . . . . . 9  |-  ( (
bday `  x )  e.  ( bday " A
)  ->  E. w  w  e.  ( bday " A ) )
119, 10syl6 29 . . . . . . . 8  |-  ( A 
C_  No  ->  ( x  e.  A  ->  E. w  w  e.  ( bday " A ) ) )
1211exlimdv 1636 . . . . . . 7  |-  ( A 
C_  No  ->  ( E. x  x  e.  A  ->  E. w  w  e.  ( bday " A
) ) )
13 n0 3540 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
14 n0 3540 . . . . . . 7  |-  ( (
bday " A )  =/=  (/) 
<->  E. w  w  e.  ( bday " A
) )
1512, 13, 143imtr4g 261 . . . . . 6  |-  ( A 
C_  No  ->  ( A  =/=  (/)  ->  ( bday " A )  =/=  (/) ) )
1615impcom 419 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( bday " A )  =/=  (/) )
17 onint 4665 . . . . 5  |-  ( ( ( bday " A
)  C_  On  /\  ( bday " A )  =/=  (/) )  ->  |^| ( bday " A )  e.  ( bday " A
) )
183, 16, 17sylancr 644 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  |^| ( bday " A )  e.  ( bday " A
) )
19 bdayfn 24891 . . . . . 6  |-  bday  Fn  No
20 fvelimab 5658 . . . . . 6  |-  ( (
bday  Fn  No  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2119, 20mpan 651 . . . . 5  |-  ( A 
C_  No  ->  ( |^| ( bday " A )  e.  ( bday " A
)  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A
) ) )
2221adantl 452 . . . 4  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  ( |^| ( bday " A
)  e.  ( bday " A )  <->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) ) )
2318, 22mpbid 201 . . 3  |-  ( ( A  =/=  (/)  /\  A  C_  No )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
24233adant3 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
) )  ->  E. w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
25 ssel 3250 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( w  e.  A  ->  w  e.  No ) )
26 ssel 3250 . . . . . . . . 9  |-  ( A 
C_  No  ->  ( t  e.  A  ->  t  e.  No ) )
2725, 26anim12d 546 . . . . . . . 8  |-  ( A 
C_  No  ->  ( ( w  e.  A  /\  t  e.  A )  ->  ( w  e.  No  /\  t  e.  No ) ) )
2827imp 418 . . . . . . 7  |-  ( ( A  C_  No  /\  (
w  e.  A  /\  t  e.  A )
)  ->  ( w  e.  No  /\  t  e.  No ) )
2928ad2ant2r 727 . . . . . 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 24902 . . . . . . 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 418 . . . . . 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 437 . . . . . . . . 9  |-  ( ( w  e.  A  /\  t  e.  A )  <->  ( t  e.  A  /\  w  e.  A )
)
33 ancom 437 . . . . . . . . 9  |-  ( ( ( bday `  w
)  =  |^| ( bday " A )  /\  ( bday `  t )  =  |^| ( bday " A
) )  <->  ( ( bday `  t )  = 
|^| ( bday " A
)  /\  ( bday `  w )  =  |^| ( bday " A ) ) )
3432, 33anbi12i 678 . . . . . . . 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 24902 . . . . . . . 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 208 . . . . . . 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 418 . . . . . 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 24886 . . . . . . 7  |-  ( ( w  e.  No  /\  t  e.  No )  ->  ( w  =  t  <-> 
( -.  w <
s t  /\  -.  t < s w ) ) )
3938biimpar 471 . . . . . 6  |-  ( ( ( w  e.  No  /\  t  e.  No )  /\  ( -.  w < s t  /\  -.  t < s w ) )  ->  w  =  t )
4029, 31, 37, 39syl12anc 1180 . . . . 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 588 . . . 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 2710 . . 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 973 . 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 5605 . . . 4  |-  ( w  =  t  ->  ( bday `  w )  =  ( bday `  t
) )
4544eqeq1d 2366 . . 3  |-  ( w  =  t  ->  (
( bday `  w )  =  |^| ( bday " A
)  <->  ( bday `  t
)  =  |^| ( bday " A ) ) )
4645reu4 3035 . 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 645 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 176    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   E!wreu 2621    C_ wss 3228   (/)c0 3531   |^|cint 3941   class class class wbr 4102   Oncon0 4471   dom cdm 4768   ran crn 4769   "cima 4771   Fun wfun 5328    Fn wfn 5329   ` cfv 5334   Nocsur 24852   < scslt 24853   bdaycbday 24854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1o 6563  df-2o 6564  df-no 24855  df-slt 24856  df-bday 24857
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