Theorem nocvxmin 31894

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.

Step	Hyp	Ref	Expression
1		imassrn 5477	. . . . . 6 |- ( bday " A ) C_ ran bday
2		bdayrn 31891	. . . . . 6 |- ran bday = On
3	1, 2	sseqtri 3637	. . . . 5 |- ( bday " A ) C_ On
4		bdaydm 31890	. . . . . . . . . . 11 |- dom bday = No
5	4	sseq2i 3630	. . . . . . . . . 10 |- ( A C_ dom bday <-> A C_ No )
6		bdayfun 31888	. . . . . . . . . . 11 |- Fun bday
7		funfvima2 6493	. . . . . . . . . . 11 |- ( ( Fun bday /\ A C_ dom bday ) -> ( x e. A -> ( bday ` x ) e. ( bday " A ) ) )
8	6, 7	mpan 706	. . . . . . . . . 10 |- ( A C_ dom bday -> ( x e. A -> ( bday ` x ) e. ( bday " A ) ) )
9	5, 8	sylbir 225	. . . . . . . . 9 |- ( A C_ No -> ( x e. A -> ( bday ` x ) e. ( bday " A ) ) )
10		elex2 3216	. . . . . . . . 9 |- ( ( bday ` x ) e. ( bday " A ) -> E. w w e. ( bday " A ) )
11	9, 10	syl6 35	. . . . . . . 8 |- ( A C_ No -> ( x e. A -> E. w w e. ( bday " A ) ) )
12	11	exlimdv 1861	. . . . . . 7 |- ( A C_ No -> ( E. x x e. A -> E. w w e. ( bday " A ) ) )
13		n0 3931	. . . . . . 7 |- ( A =/= (/) <-> E. x x e. A )
14		n0 3931	. . . . . . 7 |- ( ( bday " A ) =/= (/) <-> E. w w e. ( bday " A ) )
15	12, 13, 14	3imtr4g 285	. . . . . 6 |- ( A C_ No -> ( A =/= (/) -> ( bday " A ) =/= (/) ) )
16	15	impcom 446	. . . . 5 |- ( ( A =/= (/) /\ A C_ No ) -> ( bday " A ) =/= (/) )
17		onint 6995	. . . . 5 |- ( ( ( bday " A ) C_ On /\ ( bday " A ) =/= (/) ) -> |^| ( bday " A ) e. ( bday " A ) )
18	3, 16, 17	sylancr 695	. . . 4 |- ( ( A =/= (/) /\ A C_ No ) -> |^| ( bday " A ) e. ( bday " A ) )
19		bdayfn 31889	. . . . . 6 |- bday Fn No
20		fvelimab 6253	. . . . . 6 |- ( ( bday Fn No /\ A C_ No ) -> ( |^| ( bday " A ) e. ( bday " A ) <-> E. w e. A ( bday ` w ) = |^| ( bday " A ) ) )
21	19, 20	mpan 706	. . . . 5 |- ( A C_ No -> ( |^| ( bday " A ) e. ( bday " A ) <-> E. w e. A ( bday ` w ) = |^| ( bday " A ) ) )
22	21	adantl 482	. . . 4 |- ( ( A =/= (/) /\ A C_ No ) -> ( |^| ( bday " A ) e. ( bday " A ) <-> E. w e. A ( bday ` w ) = |^| ( bday " A ) ) )
23	18, 22	mpbid 222	. . 3 |- ( ( A =/= (/) /\ A C_ No ) -> E. w e. A ( bday ` w ) = |^| ( bday " A ) )
24	23	3adant3 1081	. 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 3597	. . . . . . . . 9 |- ( A C_ No -> ( w e. A -> w e. No ) )
26		ssel 3597	. . . . . . . . 9 |- ( A C_ No -> ( t e. A -> t e. No ) )
27	25, 26	anim12d 586	. . . . . . . 8 |- ( A C_ No -> ( ( w e. A /\ t e. A ) -> ( w e. No /\ t e. No ) ) )
28	27	imp 445	. . . . . . 7 |- ( ( A C_ No /\ ( w e. A /\ t e. A ) ) -> ( w e. No /\ t e. No ) )
29	28	ad2ant2r 783	. . . . . 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 31893	. . . . . . 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 ) )
31	30	imp 445	. . . . . 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 466	. . . . . . . . 9 |- ( ( w e. A /\ t e. A ) <-> ( t e. A /\ w e. A ) )
33		ancom 466	. . . . . . . . 9 |- ( ( ( bday ` w ) = |^| ( bday " A ) /\ ( bday ` t ) = |^| ( bday " A ) ) <-> ( ( bday ` t ) = |^| ( bday " A ) /\ ( bday ` w ) = |^| ( bday " A ) ) )
34	32, 33	anbi12i 733	. . . . . . . 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 31893	. . . . . . . 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 ) )
36	34, 35	syl5bi 232	. . . . . . 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 ) )
37	36	imp 445	. . . . . 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 31875	. . . . . . 7 |- ( ( w e. No /\ t e. No ) -> ( w = t <-> ( -. w <s t /\ -. t <s w ) ) )
39	38	biimpar 502	. . . . . 6 |- ( ( ( w e. No /\ t e. No ) /\ ( -. w <s t /\ -. t <s w ) ) -> w = t )
40	29, 31, 37, 39	syl12anc 1324	. . . . 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 )
41	40	exp32 631	. . . 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 ) ) )
42	41	ralrimivv 2970	. . 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 ) )
43	42	3adant1 1079	. 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 6191	. . . 4 |- ( w = t -> ( bday ` w ) = ( bday ` t ) )
45	44	eqeq1d 2624	. . 3 |- ( w = t -> ( ( bday ` w ) = |^| ( bday " A ) <-> ( bday ` t ) = |^| ( bday " A ) ) )
46	45	reu4 3400	. 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 ) ) )
47	24, 43, 46	sylanbrc 698	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 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   C_ wss 3574   (/)c0 3915   |^|cint 4475   class class class wbr 4653   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   Nocsur 31793   <scslt 31794   bdaycbday 31795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798

This theorem is referenced by:  conway  31910