Theorem sltsgn1 29661

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 29656	. 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 5858	. . . 4 |- ( A ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
3		fvex 5858	. . . 4 |- ( B ` |^| { k e. On | ( A ` k ) =/= ( B ` k ) } ) e. _V
4	2, 3	brtp 29419	. . 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 382	. . . . 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 463	. . . 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 463	. . . 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 383	. . . . 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 463	. . . 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 1289	. . 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 366   /\ wa 367   \/ w3o 970   = wceq 1398   e. wcel 1823   =/= wne 2649   {crab 2808   (/)c0 3783   {ctp 4020   <.cop 4022   |^|cint 4271   class class class wbr 4439   Oncon0 4867   ` cfv 5570   1oc1o 7115   2oc2o 7116   Nocsur 29640   <scslt 29641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-iota 5534  df-fv 5578  df-1o 7122  df-2o 7123  df-slt 29644

This theorem is referenced by: (None)