Theorem sltsgn1 27771

Description: If A <s B, then the sign of A at the first place they differ is either undefined or 1o (Contributed by Scott Fenton, 4-Sep-2011.)

Assertion

Ref	Expression
sltsgn1	|- ( ( A e. No /\ B e. No ) -> ( A <s B -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) ) )

Distinct variable groups:   A, k   B, k

Proof of Theorem sltsgn1

Step	Hyp	Ref	Expression
1		sltval2 27766	. 2 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) ) )
2		fvex 5696	. . . 4 |- ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
3		fvex 5696	. . . 4 |- ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
4	2, 3	brtp 27528	. . 3 |- ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) <-> ( ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) ) \/ ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) \/ ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) ) )
5		olc 384	. . . . 5 |- ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
6	5	adantr 465	. . . 4 |- ( ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) ) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
7	5	adantr 465	. . . 4 |- ( ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
8		orc 385	. . . . 5 |- ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
9	8	adantr 465	. . . 4 |- ( ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
10	6, 7, 9	3jaoi 1281	. . 3 |- ( ( ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) ) \/ ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) \/ ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) /\ ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 2o ) ) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
11	4, 10	sylbi 195	. 2 |- ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) )
12	1, 11	syl6bi 228	1 |- ( ( A e. No /\ B e. No ) -> ( A <s B -> ( ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = (/) \/ ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) = 1o ) ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   \/ w3o 964   = wceq 1369   e. wcel 1756   =/= wne 2601   {crab 2714   (/)c0 3632   {ctp 3876   <.cop 3878   |^|cint 4123   class class class wbr 4287   Oncon0 4714   ` cfv 5413   1oc1o 6905   2oc2o 6906   Nocsur 27750   <scslt 27751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-iota 5376  df-fv 5421  df-1o 6912  df-2o 6913  df-slt 27754

This theorem is referenced by: (None)