Theorem nocvxmin 14029

Description: Given a non-empty convex class of surreals, there is a unique birthday-minimal element of that class.

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

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (w = t -> ( bday ` w) = ( bday ` t))
2	1	eqeq1d 1892	. . 3 |- (w = t -> (( bday ` w) = |^|( bday "A) <-> ( bday ` t) = |^|( bday "A)))
3	2	reu4 2446	. 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)))
4		onint 3876	. . . . 5 |- ((( bday "A) C_ On /\ ( bday "A) =/= (/)) -> |^|( bday "A) e. ( bday "A))
5		imassrn 4278	. . . . . 6 |- ( bday "A) C_ ran bday
6		bdayrn 14014	. . . . . 6 |- ran bday = On
7	5, 6	sseqtri 2649	. . . . 5 |- ( bday "A) C_ On
8		bdaydm 14015	. . . . . . . . . . 11 |- dom bday = No
9	8	sseq2i 2642	. . . . . . . . . 10 |- (A C_ dom bday <-> A C_ No )
10		bdayfun 14013	. . . . . . . . . . 11 |- Fun bday
11		funfvima2 4829	. . . . . . . . . . 11 |- ((Fun bday /\ A C_ dom bday ) -> (x e. A -> ( bday ` x) e. ( bday "A)))
12	10, 11	mpan 759	. . . . . . . . . 10 |- (A C_ dom bday -> (x e. A -> ( bday ` x) e. ( bday "A)))
13	9, 12	sylbir 218	. . . . . . . . 9 |- (A C_ No -> (x e. A -> ( bday ` x) e. ( bday "A)))
14		fvex 4689	. . . . . . . . . 10 |- ( bday ` x) e. _V
15		eleq1 1957	. . . . . . . . . 10 |- (w = ( bday ` x) -> (w e. ( bday "A) <-> ( bday ` x) e. ( bday "A)))
16	14, 15	cla4ev 2371	. . . . . . . . 9 |- (( bday ` x) e. ( bday "A) -> E.w w e. ( bday "A))
17	13, 16	syl6 25	. . . . . . . 8 |- (A C_ No -> (x e. A -> E.w w e. ( bday "A)))
18	17	19.23adv 1584	. . . . . . 7 |- (A C_ No -> (E.x x e. A -> E.w w e. ( bday "A)))
19		n0 2884	. . . . . . 7 |- (A =/= (/) <-> E.x x e. A)
20		n0 2884	. . . . . . 7 |- (( bday "A) =/= (/) <-> E.w w e. ( bday "A))
21	18, 19, 20	3imtr4g 612	. . . . . 6 |- (A C_ No -> (A =/= (/) -> ( bday "A) =/= (/)))
22	21	impcom 378	. . . . 5 |- ((A =/= (/) /\ A C_ No ) -> ( bday "A) =/= (/))
23	4, 7, 22	sylancr 526	. . . 4 |- ((A =/= (/) /\ A C_ No ) -> |^|( bday "A) e. ( bday "A))
24		bdayfn 14016	. . . . . 6 |- bday Fn No
25		fvelimab 4725	. . . . . 6 |- (( bday Fn No /\ A C_ No ) -> (|^|( bday "A) e. ( bday "A) <-> E.w e. A ( bday ` w) = |^|( bday "A)))
26	24, 25	mpan 759	. . . . 5 |- (A C_ No -> (|^|( bday "A) e. ( bday "A) <-> E.w e. A ( bday ` w) = |^|( bday "A)))
27	26	adantl 424	. . . 4 |- ((A =/= (/) /\ A C_ No ) -> (|^|( bday "A) e. ( bday "A) <-> E.w e. A ( bday ` w) = |^|( bday "A)))
28	23, 27	mpbid 212	. . 3 |- ((A =/= (/) /\ A C_ No ) -> E.w e. A ( bday ` w) = |^|( bday "A))
29	28	3adant3 896	. 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))
30		ssel 2615	. . . . . . . . 9 |- (A C_ No -> (w e. A -> w e. No ))
31		ssel 2615	. . . . . . . . 9 |- (A C_ No -> (t e. A -> t e. No ))
32	30, 31	anim12d 617	. . . . . . . 8 |- (A C_ No -> ((w e. A /\ t e. A) -> (w e. No /\ t e. No )))
33	32	imp 377	. . . . . . 7 |- ((A C_ No /\ (w e. A /\ t e. A)) -> (w e. No /\ t e. No ))
34	33	ad2ant2r 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 e. No /\ t e. No ))
35		nocvxminlem 14028	. . . . . . 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))
36	35	imp 377	. . . . . 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)
37		nocvxminlem 14028	. . . . . . . 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))
38		ancom 482	. . . . . . . . 9 |- ((w e. A /\ t e. A) <-> (t e. A /\ w e. A))
39		ancom 482	. . . . . . . . 9 |- ((( bday ` w) = |^|( bday "A) /\ ( bday ` t) = |^|( bday "A)) <-> (( bday ` t) = |^|( bday "A) /\ ( bday ` w) = |^|( bday "A)))
40	38, 39	anbi12i 540	. . . . . . . 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))))
41	37, 40	syl5ib 223	. . . . . . 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))
42	41	imp 377	. . . . . 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)
43		axsltso 14007	. . . . . . . 8 |- <s Or No
44		sotrieq2 3618	. . . . . . . 8 |- (( <s Or No /\ (w e. No /\ t e. No )) -> (w = t <-> (-. w <s t /\ -. t <s w)))
45	43, 44	mpan 759	. . . . . . 7 |- ((w e. No /\ t e. No ) -> (w = t <-> (-. w <s t /\ -. t <s w)))
46	45	biimpar 461	. . . . . 6 |- (((w e. No /\ t e. No ) /\ (-. w <s t /\ -. t <s w)) -> w = t)
47	34, 36, 42, 46	syl12anc 1098	. . . . 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)
48	47	exp32 408	. . . 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)))
49	48	r19.21aivv 2183	. . 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))
50	49	3adant1 894	. 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))
51	3, 29, 50	sylanbrc 527	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 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  E!wreu 2107   C_ wss 2593  (/)c0 2875  |^|cint 3214   class class class wbr 3338   Or wor 3590  Oncon0 3657  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998   No csur 13981   <s cslt 13982   bday cbday 13983

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-mpt 5006  df-1o 5177  df-2o 5178  df-no 13984  df-slt 13985  df-bday 13986

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