Theorem nosep1o 31832

Description: If the value of a surreal at a separator is 1o then the surreal is lesser. (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
nosep1o	|- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> A <s B )

Distinct variable groups:   x, A   x, B

Proof of Theorem nosep1o

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o )
2		nosepne 31831	. . . . . . . . . . . 12 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
3	2	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
4	1, 3	eqnetrrd 2862	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> 1o =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
5	4	necomd 2849	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 1o )
6	5	neneqd 2799	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> -. ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o )
7		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> B e. No )
8		nofv 31810	. . . . . . . . 9 |- ( B e. No -> ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) )
9	7, 8	syl 17	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) )
10		3orel2 31592	. . . . . . . 8 |- ( -. ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o -> ( ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
11	6, 9, 10	sylc 65	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) )
12		eqid 2622	. . . . . . 7 |- 1o = 1o
13	11, 12	jctil 560	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( 1o = 1o /\ ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
14		andi 911	. . . . . 6 |- ( ( 1o = 1o /\ ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) \/ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) <-> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
15	13, 14	sylib 208	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
16		3mix1 1230	. . . . . 6 |- ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) -> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( 1o = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
17		3mix2 1231	. . . . . 6 |- ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( 1o = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
18	16, 17	jaoi 394	. . . . 5 |- ( ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( 1o = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
19	15, 18	syl 17	. . . 4 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( 1o = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
20		1on 7567	. . . . . 6 |- 1o e. On
21	20	elexi 3213	. . . . 5 |- 1o e. _V
22		fvex 6201	. . . . 5 |- ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) e. _V
23	21, 22	brtp 31639	. . . 4 |- ( 1o { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> ( ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( 1o = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( 1o = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
24	19, 23	sylibr 224	. . 3 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> 1o { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
25	1, 24	eqbrtrd 4675	. 2 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
26		simpl1 1064	. . 3 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> A e. No )
27		sltval2 31809	. . 3 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) )
28	26, 7, 27	syl2anc 693	. 2 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> ( A <s B <-> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) )
29	25, 28	mpbird 247	1 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> A <s B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   (/)c0 3915   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   Oncon0 5723   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

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

This theorem is referenced by:  noetalem3  31865