Theorem nocvxminlem 14028

Description: Lemma for nocvxmin 14029. Given two birthday-minimal elements of a convex class of surreals, they are not comparable.

Assertion

Ref	Expression
nocvxminlem	|- ((A C_ No /\ A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A)) -> (((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A))) -> -. X <s Y))

Distinct variable groups:   x,A,y,z   x,X,y,z   y,Y,z

Proof of Theorem nocvxminlem

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . . . . . . . . . . 14 |- (x = X -> (x <s z <-> X <s z))
2	1	anbi1d 679	. . . . . . . . . . . . 13 |- (x = X -> ((x <s z /\ z <s y) <-> (X <s z /\ z <s y)))
3	2	imbi1d 675	. . . . . . . . . . . 12 |- (x = X -> (((x <s z /\ z <s y) -> z e. A) <-> ((X <s z /\ z <s y) -> z e. A)))
4	3	ralbidv 2123	. . . . . . . . . . 11 |- (x = X -> (A.z e. No ((x <s z /\ z <s y) -> z e. A) <-> A.z e. No ((X <s z /\ z <s y) -> z e. A)))
5		breq2 3342	. . . . . . . . . . . . . 14 |- (y = Y -> (z <s y <-> z <s Y))
6	5	anbi2d 678	. . . . . . . . . . . . 13 |- (y = Y -> ((X <s z /\ z <s y) <-> (X <s z /\ z <s Y)))
7	6	imbi1d 675	. . . . . . . . . . . 12 |- (y = Y -> (((X <s z /\ z <s y) -> z e. A) <-> ((X <s z /\ z <s Y) -> z e. A)))
8	7	ralbidv 2123	. . . . . . . . . . 11 |- (y = Y -> (A.z e. No ((X <s z /\ z <s y) -> z e. A) <-> A.z e. No ((X <s z /\ z <s Y) -> z e. A)))
9	4, 8	rcla42v 2384	. . . . . . . . . 10 |- ((X e. A /\ Y e. A) -> (A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A) -> A.z e. No ((X <s z /\ z <s Y) -> z e. A)))
10		breq2 3342	. . . . . . . . . . . . . . . 16 |- (z = w -> (X <s z <-> X <s w))
11		breq1 3341	. . . . . . . . . . . . . . . 16 |- (z = w -> (z <s Y <-> w <s Y))
12	10, 11	anbi12d 690	. . . . . . . . . . . . . . 15 |- (z = w -> ((X <s z /\ z <s Y) <-> (X <s w /\ w <s Y)))
13		eleq1 1957	. . . . . . . . . . . . . . 15 |- (z = w -> (z e. A <-> w e. A))
14	12, 13	imbi12d 688	. . . . . . . . . . . . . 14 |- (z = w -> (((X <s z /\ z <s Y) -> z e. A) <-> ((X <s w /\ w <s Y) -> w e. A)))
15	14	rcla4v 2376	. . . . . . . . . . . . 13 |- (w e. No -> (A.z e. No ((X <s z /\ z <s Y) -> z e. A) -> ((X <s w /\ w <s Y) -> w e. A)))
16		bdaydm 14015	. . . . . . . . . . . . . . . . . . . . . 22 |- dom bday = No
17	16	sseq2i 2642	. . . . . . . . . . . . . . . . . . . . 21 |- (A C_ dom bday <-> A C_ No )
18		bdayfun 14013	. . . . . . . . . . . . . . . . . . . . . 22 |- Fun bday
19		funfvima2 4829	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Fun bday /\ A C_ dom bday ) -> (w e. A -> ( bday ` w) e. ( bday "A)))
20	18, 19	mpan 759	. . . . . . . . . . . . . . . . . . . . 21 |- (A C_ dom bday -> (w e. A -> ( bday ` w) e. ( bday "A)))
21	17, 20	sylbir 218	. . . . . . . . . . . . . . . . . . . 20 |- (A C_ No -> (w e. A -> ( bday ` w) e. ( bday "A)))
22	21	imp 377	. . . . . . . . . . . . . . . . . . 19 |- ((A C_ No /\ w e. A) -> ( bday ` w) e. ( bday "A))
23		intss1 3231	. . . . . . . . . . . . . . . . . . 19 |- (( bday ` w) e. ( bday "A) -> |^|( bday "A) C_ ( bday ` w))
24	22, 23	syl 12	. . . . . . . . . . . . . . . . . 18 |- ((A C_ No /\ w e. A) -> |^|( bday "A) C_ ( bday ` w))
25		ontri1 3695	. . . . . . . . . . . . . . . . . . 19 |- ((|^|( bday "A) e. On /\ ( bday ` w) e. On) -> (|^|( bday "A) C_ ( bday ` w) <-> -. ( bday ` w) e. |^|( bday "A)))
26		oninton 3881	. . . . . . . . . . . . . . . . . . . 20 |- ((( bday "A) C_ On /\ ( bday "A) =/= (/)) -> |^|( bday "A) e. On)
27		imassrn 4278	. . . . . . . . . . . . . . . . . . . . 21 |- ( bday "A) C_ ran bday
28		bdayrn 14014	. . . . . . . . . . . . . . . . . . . . 21 |- ran bday = On
29	27, 28	sseqtri 2649	. . . . . . . . . . . . . . . . . . . 20 |- ( bday "A) C_ On
30		ne0i 2881	. . . . . . . . . . . . . . . . . . . . 21 |- (( bday ` w) e. ( bday "A) -> ( bday "A) =/= (/))
31	22, 30	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- ((A C_ No /\ w e. A) -> ( bday "A) =/= (/))
32	26, 29, 31	sylancr 526	. . . . . . . . . . . . . . . . . . 19 |- ((A C_ No /\ w e. A) -> |^|( bday "A) e. On)
33		bdayelon 14017	. . . . . . . . . . . . . . . . . . 19 |- ( bday ` w) e. On
34	25, 32, 33	sylancl 525	. . . . . . . . . . . . . . . . . 18 |- ((A C_ No /\ w e. A) -> (|^|( bday "A) C_ ( bday ` w) <-> -. ( bday ` w) e. |^|( bday "A)))
35	24, 34	mpbid 212	. . . . . . . . . . . . . . . . 17 |- ((A C_ No /\ w e. A) -> -. ( bday ` w) e. |^|( bday "A))
36	35	ex 402	. . . . . . . . . . . . . . . 16 |- (A C_ No -> (w e. A -> -. ( bday ` w) e. |^|( bday "A)))
37		eleq2 1958	. . . . . . . . . . . . . . . . . 18 |- (( bday ` X) = |^|( bday "A) -> (( bday ` w) e. ( bday ` X) <-> ( bday ` w) e. |^|( bday "A)))
38	37	notbid 673	. . . . . . . . . . . . . . . . 17 |- (( bday ` X) = |^|( bday "A) -> (-. ( bday ` w) e. ( bday ` X) <-> -. ( bday ` w) e. |^|( bday "A)))
39	38	biimprcd 173	. . . . . . . . . . . . . . . 16 |- (-. ( bday ` w) e. |^|( bday "A) -> (( bday ` X) = |^|( bday "A) -> -. ( bday ` w) e. ( bday ` X)))
40	36, 39	syl6 25	. . . . . . . . . . . . . . 15 |- (A C_ No -> (w e. A -> (( bday ` X) = |^|( bday "A) -> -. ( bday ` w) e. ( bday ` X))))
41	40	com3l 38	. . . . . . . . . . . . . 14 |- (w e. A -> (( bday ` X) = |^|( bday "A) -> (A C_ No -> -. ( bday ` w) e. ( bday ` X))))
42	41	adantrd 427	. . . . . . . . . . . . 13 |- (w e. A -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (A C_ No -> -. ( bday ` w) e. ( bday ` X))))
43	15, 42	syl8 27	. . . . . . . . . . . 12 |- (w e. No -> (A.z e. No ((X <s z /\ z <s Y) -> z e. A) -> ((X <s w /\ w <s Y) -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (A C_ No -> -. ( bday ` w) e. ( bday ` X))))))
44	43	com35 47	. . . . . . . . . . 11 |- (w e. No -> (A.z e. No ((X <s z /\ z <s Y) -> z e. A) -> (A C_ No -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X))))))
45	44	com4l 43	. . . . . . . . . 10 |- (A.z e. No ((X <s z /\ z <s Y) -> z e. A) -> (A C_ No -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (w e. No -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X))))))
46	9, 45	syl6 25	. . . . . . . . 9 |- ((X e. A /\ Y e. A) -> (A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A) -> (A C_ No -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (w e. No -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X)))))))
47	46	com3l 38	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A) -> (A C_ No -> ((X e. A /\ Y e. A) -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (w e. No -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X)))))))
48	47	impcom 378	. . . . . . 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)) -> ((X e. A /\ Y e. A) -> ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> (w e. No -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X))))))
49	48	imp42 396	. . . . . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) /\ w e. No ) -> ((X <s w /\ w <s Y) -> -. ( bday ` w) e. ( bday ` X)))
50	49	con2d 107	. . . . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) /\ w e. No ) -> (( bday ` w) e. ( bday ` X) -> -. (X <s w /\ w <s Y)))
51		3anass 862	. . . . . . 7 |- ((( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y) <-> (( bday ` w) e. ( bday ` X) /\ (X <s w /\ w <s Y)))
52	51	notbii 204	. . . . . 6 |- (-. (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y) <-> -. (( bday ` w) e. ( bday ` X) /\ (X <s w /\ w <s Y)))
53		imnan 261	. . . . . 6 |- ((( bday ` w) e. ( bday ` X) -> -. (X <s w /\ w <s Y)) <-> -. (( bday ` w) e. ( bday ` X) /\ (X <s w /\ w <s Y)))
54	52, 53	bitr4i 193	. . . . 5 |- (-. (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y) <-> (( bday ` w) e. ( bday ` X) -> -. (X <s w /\ w <s Y)))
55	50, 54	sylibr 217	. . . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) /\ w e. No ) -> -. (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y))
56	55	nrexdv 2193	. . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) -> -. E.w e. No (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y))
57		axdense 14027	. . . . 5 |- (((X e. No /\ Y e. No ) /\ (( bday ` X) = ( bday ` Y) /\ X <s Y)) -> E.w e. No (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y))
58	57	anassrs 489	. . . 4 |- ((((X e. No /\ Y e. No ) /\ ( bday ` X) = ( bday ` Y)) /\ X <s Y) -> E.w e. No (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y))
59		ssel 2615	. . . . . . . . 9 |- (A C_ No -> (X e. A -> X e. No ))
60		ssel 2615	. . . . . . . . 9 |- (A C_ No -> (Y e. A -> Y e. No ))
61	59, 60	anim12d 617	. . . . . . . 8 |- (A C_ No -> ((X e. A /\ Y e. A) -> (X e. No /\ Y e. No )))
62	61	imp 377	. . . . . . 7 |- ((A C_ No /\ (X e. A /\ Y e. A)) -> (X e. No /\ Y e. No ))
63		eqtr3 1907	. . . . . . 7 |- ((( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)) -> ( bday ` X) = ( bday ` Y))
64	62, 63	anim12i 360	. . . . . 6 |- (((A C_ No /\ (X e. A /\ Y e. A)) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A))) -> ((X e. No /\ Y e. No ) /\ ( bday ` X) = ( bday ` Y)))
65	64	anasss 488	. . . . 5 |- ((A C_ No /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) -> ((X e. No /\ Y e. No ) /\ ( bday ` X) = ( bday ` Y)))
66	65	adantlr 429	. . . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) -> ((X e. No /\ Y e. No ) /\ ( bday ` X) = ( bday ` Y)))
67	58, 66	sylan 497	. . 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)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) /\ X <s Y) -> E.w e. No (( bday ` w) e. ( bday ` X) /\ X <s w /\ w <s Y))
68	56, 67	mtand 520	. 2 |- (((A C_ No /\ A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A)) /\ ((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A)))) -> -. X <s Y)
69	68	ex 402	1 |- ((A C_ No /\ A.x e. A A.y e. A A.z e. No ((x <s z /\ z <s y) -> z e. A)) -> (((X e. A /\ Y e. A) /\ (( bday ` X) = |^|( bday "A) /\ ( bday ` Y) = |^|( bday "A))) -> -. X <s Y))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  |^|cint 3214   class class class wbr 3338  Oncon0 3657  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992  ` cfv 3998   No csur 13981   <s cslt 13982   bday cbday 13983

This theorem is referenced by:  nocvxmin 14029

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-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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