Theorem nosepdm 31834

Description: The first place two surreals differ is an element of the larger of their domains. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem nosepdm

Step	Hyp	Ref	Expression
1		sltso 31827	. . . 4 |- <s Or No
2		sotrine 31658	. . . 4 |- ( ( <s Or No /\ ( A e. No /\ B e. No ) ) -> ( A =/= B <-> ( A <s B \/ B <s A ) ) )
3	1, 2	mpan 706	. . 3 |- ( ( A e. No /\ B e. No ) -> ( A =/= B <-> ( A <s B \/ B <s A ) ) )
4		nosepdmlem 31833	. . . . . 6 |- ( ( A e. No /\ B e. No /\ A <s B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
5	4	3expa 1265	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
6		simplr 792	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ B <s A ) -> B e. No )
7		simpll 790	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ B <s A ) -> A e. No )
8		simpr 477	. . . . . . 7 |- ( ( ( A e. No /\ B e. No ) /\ B <s A ) -> B <s A )
9		nosepdmlem 31833	. . . . . . 7 |- ( ( B e. No /\ A e. No /\ B <s A ) -> |^| { x e. On | ( B ` x ) =/= ( A ` x ) } e. ( dom B u. dom A ) )
10	6, 7, 8, 9	syl3anc 1326	. . . . . 6 |- ( ( ( A e. No /\ B e. No ) /\ B <s A ) -> |^| { x e. On | ( B ` x ) =/= ( A ` x ) } e. ( dom B u. dom A ) )
11		necom 2847	. . . . . . . . 9 |- ( ( A ` x ) =/= ( B ` x ) <-> ( B ` x ) =/= ( A ` x ) )
12	11	a1i 11	. . . . . . . 8 |- ( x e. On -> ( ( A ` x ) =/= ( B ` x ) <-> ( B ` x ) =/= ( A ` x ) ) )
13	12	rabbiia 3185	. . . . . . 7 |- { x e. On | ( A ` x ) =/= ( B ` x ) } = { x e. On | ( B ` x ) =/= ( A ` x ) }
14	13	inteqi 4479	. . . . . 6 |- |^| { x e. On | ( A ` x ) =/= ( B ` x ) } = |^| { x e. On | ( B ` x ) =/= ( A ` x ) }
15		uncom 3757	. . . . . 6 |- ( dom A u. dom B ) = ( dom B u. dom A )
16	10, 14, 15	3eltr4g 2718	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ B <s A ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
17	5, 16	jaodan 826	. . . 4 |- ( ( ( A e. No /\ B e. No ) /\ ( A <s B \/ B <s A ) ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
18	17	ex 450	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( A <s B \/ B <s A ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) )
19	3, 18	sylbid 230	. 2 |- ( ( A e. No /\ B e. No ) -> ( A =/= B -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) )
20	19	3impia 1261	1 |- ( ( A e. No /\ B e. No /\ A =/= B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   {crab 2916   u. cun 3572   |^|cint 4475   class class class wbr 4653   Or wor 5034   dom cdm 5114   Oncon0 5723   ` cfv 5888   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-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-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-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-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797

This theorem is referenced by:  nodenselem5  31838  noresle  31846