Theorem sltsgn1 27945

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 27940	. 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 5808	. . . 4 |- ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
3		fvex 5808	. . . 4 |- ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
4	2, 3	brtp 27702	. . 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 1282	. . 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 1370   e. wcel 1758   =/= wne 2647   {crab 2802   (/)c0 3744   {ctp 3988   <.cop 3990   |^|cint 4235   class class class wbr 4399   Oncon0 4826   ` cfv 5525   1oc1o 7022   2oc2o 7023   Nocsur 27924   <scslt 27925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-br 4400  df-opab 4458  df-tr 4493  df-eprel 4739  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-suc 4832  df-iota 5488  df-fv 5533  df-1o 7029  df-2o 7030  df-slt 27928

This theorem is referenced by: (None)