Theorem sltsgn2 14003

Description: If A <s B, then the sign of B at the first place they differ is either undefined or 2o

Assertion

Ref	Expression
sltsgn2	|- ((A e. No /\ B e. No ) -> (A <s B -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o)))

Distinct variable groups:   A,k   B,k

Proof of Theorem sltsgn2

Step	Hyp	Ref	Expression
1		sltval2 13997	. 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 4689	. . . 4 |- (A` |^|{k e. On | (A` k) =/= (B` k)}) e. _V
3		fvex 4689	. . . 4 |- (B` |^|{k e. On | (A` k) =/= (B` k)}) e. _V
4		0ex 3446	. . . 4 |- (/) e. _V
5		2on 5183	. . . . 5 |- 2o e. On
6	5	elisseti 2301	. . . 4 |- 2o e. _V
7	2, 3, 4, 6, 6	brtp 13830	. . 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)))
8		orc 291	. . . . 5 |- ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
9	8	adantl 424	. . . 4 |- (((A` |^|{k e. On | (A` k) =/= (B` k)}) = 1o /\ (B` |^|{k e. On | (A` k) =/= (B` k)}) = (/)) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
10		olc 290	. . . . 5 |- ((B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
11	10	adantl 424	. . . 4 |- (((A` |^|{k e. On | (A` k) =/= (B` k)}) = 1o /\ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
12	10	adantl 424	. . . 4 |- (((A` |^|{k e. On | (A` k) =/= (B` k)}) = (/) /\ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
13	9, 11, 12	3jaoi 1160	. . 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)) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
14	7, 13	sylbi 216	. 2 |- ((A` |^|{k e. On | (A` k) =/= (B` k)}){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` |^|{k e. On | (A` k) =/= (B` k)}) -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o))
15	1, 14	syl6bi 231	1 |- ((A e. No /\ B e. No ) -> (A <s B -> ((B` |^|{k e. On | (A` k) =/= (B` k)}) = (/) \/ (B` |^|{k e. On | (A` k) =/= (B` k)}) = 2o)))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   =/= wne 2017  {crab 2108  (/)c0 2875  <.cop 3046  {ctp 3051  |^|cint 3214   class class class wbr 3338  Oncon0 3657  ` cfv 3998  1oc1o 5172  2oc2o 5173   No csur 13981   <s cslt 13982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-1o 5177  df-2o 5178  df-slt 13985

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