Theorem nosepssdm 31836

Description: Given two non-equal surreals, their separator is less than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nosepssdm	|- ( ( A e. No /\ B e. No /\ A =/= B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A )

Distinct variable groups:   x, A   x, B

Proof of Theorem nosepssdm

Step	Hyp	Ref	Expression
1		nosepne 31831	. . . 4 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
2	1	neneqd 2799	. . 3 |- ( ( 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		nodmord 31806	. . . . . . . . 9 |- ( A e. No -> Ord dom A )
4	3	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> Ord dom A )
5		ordn2lp 5743	. . . . . . . 8 |- ( Ord dom A -> -. ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } /\ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) )
6	4, 5	syl 17	. . . . . . 7 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> -. ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } /\ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) )
7		imnan 438	. . . . . . 7 |- ( ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) <-> -. ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } /\ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) )
8	6, 7	sylibr 224	. . . . . 6 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) )
9	8	imp 445	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A )
10		ndmfv 6218	. . . . 5 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
11	9, 10	syl 17	. . . 4 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
12		nosepeq 31835	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( A ` dom A ) = ( B ` dom A ) )
13		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> A e. No )
14	13, 3	syl 17	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> Ord dom A )
15		ordirr 5741	. . . . . . . . . 10 |- ( Ord dom A -> -. dom A e. dom A )
16		ndmfv 6218	. . . . . . . . . 10 |- ( -. dom A e. dom A -> ( A ` dom A ) = (/) )
17	14, 15, 16	3syl 18	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( A ` dom A ) = (/) )
18	17	eqeq1d 2624	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( ( A ` dom A ) = ( B ` dom A ) <-> (/) = ( B ` dom A ) ) )
19		eqcom 2629	. . . . . . . 8 |- ( (/) = ( B ` dom A ) <-> ( B ` dom A ) = (/) )
20	18, 19	syl6bb 276	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( ( A ` dom A ) = ( B ` dom A ) <-> ( B ` dom A ) = (/) ) )
21		simpl2 1065	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> B e. No )
22		nofun 31802	. . . . . . . . . . 11 |- ( B e. No -> Fun B )
23	21, 22	syl 17	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> Fun B )
24		nosgnn0 31811	. . . . . . . . . . 11 |- -. (/) e. { 1o , 2o }
25		norn 31804	. . . . . . . . . . . . 13 |- ( B e. No -> ran B C_ { 1o , 2o } )
26	21, 25	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ran B C_ { 1o , 2o } )
27	26	sseld 3602	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( (/) e. ran B -> (/) e. { 1o , 2o } ) )
28	24, 27	mtoi 190	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> -. (/) e. ran B )
29		funeldmb 31661	. . . . . . . . . 10 |- ( ( Fun B /\ -. (/) e. ran B ) -> ( dom A e. dom B <-> ( B ` dom A ) =/= (/) ) )
30	23, 28, 29	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( dom A e. dom B <-> ( B ` dom A ) =/= (/) ) )
31	30	necon2bbid 2837	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( ( B ` dom A ) = (/) <-> -. dom A e. dom B ) )
32		nodmord 31806	. . . . . . . . . . . 12 |- ( B e. No -> Ord dom B )
33	32	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> Ord dom B )
34		ordtr1 5767	. . . . . . . . . . 11 |- ( Ord dom B -> ( ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } /\ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) -> dom A e. dom B ) )
35	33, 34	syl 17	. . . . . . . . . 10 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> ( ( dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } /\ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) -> dom A e. dom B ) )
36	35	expdimp 453	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B -> dom A e. dom B ) )
37	36	con3d 148	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( -. dom A e. dom B -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
38	31, 37	sylbid 230	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( ( B ` dom A ) = (/) -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
39	20, 38	sylbid 230	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( ( A ` dom A ) = ( B ` dom A ) -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
40	12, 39	mpd 15	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B )
41		ndmfv 6218	. . . . 5 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
42	40, 41	syl 17	. . . 4 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
43	11, 42	eqtr4d 2659	. . 3 |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
44	2, 43	mtand 691	. 2 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> -. dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } )
45		nosepon 31818	. . 3 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On )
46		nodmon 31803	. . . 4 |- ( A e. No -> dom A e. On )
47	46	3ad2ant1 1082	. . 3 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> dom A e. On )
48		ontri1 5757	. . 3 |- ( ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On /\ dom A e. On ) -> ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A <-> -. dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
49	45, 47, 48	syl2anc 693	. 2 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A <-> -. dom A e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) )
50	44, 49	mpbird 247	1 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   (/)c0 3915   {cpr 4179   |^|cint 4475   dom cdm 5114   ran crn 5115   Ord word 5722   Oncon0 5723   Fun wfun 5882   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793

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:  nosupbnd2lem1  31861  noetalem3  31865