Theorem nodenselem8 31841

Description: Lemma for nodense 31842. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)

Assertion

Ref	Expression
nodenselem8	|- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B <-> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )

Distinct variable groups:   A, a   B, a

Proof of Theorem nodenselem8

Step	Hyp	Ref	Expression
1		nodenselem5 31838	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) )
2	1	exp32 631	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) -> ( A <s B -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) ) )
3	2	3impia 1261	. . 3 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) )
4		sltval2 31809	. . . . 5 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
5	4	3adant3 1081	. . . 4 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
6		fvex 6201	. . . . . 6 |- ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
7		fvex 6201	. . . . . 6 |- ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
8	6, 7	brtp 31639	. . . . 5 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
9		eleq2 2690	. . . . . . . . . . . . 13 |- ( ( bday ` A ) = ( bday ` B ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) <-> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) )
10	9	biimpd 219	. . . . . . . . . . . 12 |- ( ( bday ` A ) = ( bday ` B ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) )
11		nosgnn0 31811	. . . . . . . . . . . . . . 15 |- -. (/) e. { 1o , 2o }
12		nofnbday 31805	. . . . . . . . . . . . . . . . 17 |- ( B e. No -> B Fn ( bday ` B ) )
13		fnfvelrn 6356	. . . . . . . . . . . . . . . . . 18 |- ( ( B Fn ( bday ` B ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. ran B )
14		eleq1 2689	. . . . . . . . . . . . . . . . . 18 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. ran B <-> (/) e. ran B ) )
15	13, 14	syl5ibcom 235	. . . . . . . . . . . . . . . . 17 |- ( ( B Fn ( bday ` B ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. ran B ) )
16	12, 15	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( B e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. ran B ) )
17		norn 31804	. . . . . . . . . . . . . . . . . 18 |- ( B e. No -> ran B C_ { 1o , 2o } )
18	17	sseld 3602	. . . . . . . . . . . . . . . . 17 |- ( B e. No -> ( (/) e. ran B -> (/) e. { 1o , 2o } ) )
19	18	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( B e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> ( (/) e. ran B -> (/) e. { 1o , 2o } ) )
20	16, 19	syld 47	. . . . . . . . . . . . . . 15 |- ( ( B e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. { 1o , 2o } ) )
21	11, 20	mtoi 190	. . . . . . . . . . . . . 14 |- ( ( B e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
22	21	ex 450	. . . . . . . . . . . . 13 |- ( B e. No -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. No /\ B e. No ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` B ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) )
24	10, 23	syl9r 78	. . . . . . . . . . 11 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) ) )
25	24	3impia 1261	. . . . . . . . . 10 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) )
26	25	imp 445	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
27	26	intnand 962	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> -. ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) )
28		nofnbday 31805	. . . . . . . . . . . . 13 |- ( A e. No -> A Fn ( bday ` A ) )
29		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( A Fn ( bday ` A ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. ran A )
30		eleq1 2689	. . . . . . . . . . . . . 14 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. ran A <-> (/) e. ran A ) )
31	29, 30	syl5ibcom 235	. . . . . . . . . . . . 13 |- ( ( A Fn ( bday ` A ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. ran A ) )
32	28, 31	sylan 488	. . . . . . . . . . . 12 |- ( ( A e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. ran A ) )
33		norn 31804	. . . . . . . . . . . . . 14 |- ( A e. No -> ran A C_ { 1o , 2o } )
34	33	sseld 3602	. . . . . . . . . . . . 13 |- ( A e. No -> ( (/) e. ran A -> (/) e. { 1o , 2o } ) )
35	34	adantr 481	. . . . . . . . . . . 12 |- ( ( A e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( (/) e. ran A -> (/) e. { 1o , 2o } ) )
36	32, 35	syld 47	. . . . . . . . . . 11 |- ( ( A e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> (/) e. { 1o , 2o } ) )
37	11, 36	mtoi 190	. . . . . . . . . 10 |- ( ( A e. No /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
38	37	3ad2antl1 1223	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
39	38	intnanrd 963	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> -. ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) )
40		3orel13 31598	. . . . . . . 8 |- ( ( -. ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) /\ -. ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
41	27, 39, 40	syl2anc 693	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) /\ |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) -> ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
42	41	ex 450	. . . . . 6 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) ) )
43	42	com23 86	. . . . 5 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) ) )
44	8, 43	syl5bi 232	. . . 4 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) ) )
45	5, 44	sylbid 230	. . 3 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) ) )
46	3, 45	mpdd 43	. 2 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
47		3mix2 1231	. . . 4 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
48	47, 8	sylibr 224	. . 3 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) )
49	48, 5	syl5ibr 236	. 2 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> A <s B ) )
50	46, 49	impbid 202	1 |- ( ( A e. No /\ B e. No /\ ( bday ` A ) = ( bday ` B ) ) -> ( A <s B <-> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   (/)c0 3915   {cpr 4179   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   ran crn 5115   Oncon0 5723   Fn wfn 5883   ` cfv 5888   1oc1o 7553   2oc2o 7554   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-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:  nodense  31842