Theorem tltnle 29662

Description: In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 16966. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypotheses

Ref	Expression
tleile.b	|- B = ( Base ` K )
tleile.l	|- .<_ = ( le ` K )
tltnle.s	|- .< = ( lt ` K )

Assertion

Ref	Expression
tltnle	|- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> -. Y .<_ X ) )

Proof of Theorem tltnle

Step	Hyp	Ref	Expression
1		tospos 29658	. . 3 |- ( K e. Toset -> K e. Poset )
2		tleile.b	. . . 4 |- B = ( Base ` K )
3		tleile.l	. . . 4 |- .<_ = ( le ` K )
4		tltnle.s	. . . 4 |- .< = ( lt ` K )
5	2, 3, 4	pltval3 16967	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )
6	1, 5	syl3an1 1359	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )
7	2, 3	tleile 29661	. . 3 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .<_ X ) )
8		ibar 525	. . . 4 |- ( ( X .<_ Y \/ Y .<_ X ) -> ( -. Y .<_ X <-> ( ( X .<_ Y \/ Y .<_ X ) /\ -. Y .<_ X ) ) )
9		pm5.61 749	. . . 4 |- ( ( ( X .<_ Y \/ Y .<_ X ) /\ -. Y .<_ X ) <-> ( X .<_ Y /\ -. Y .<_ X ) )
10	8, 9	syl6rbb 277	. . 3 |- ( ( X .<_ Y \/ Y .<_ X ) -> ( ( X .<_ Y /\ -. Y .<_ X ) <-> -. Y .<_ X ) )
11	7, 10	syl 17	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ -. Y .<_ X ) <-> -. Y .<_ X ) )
12	6, 11	bitrd 268	1 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> -. Y .<_ X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   ltcplt 16941  Tosetctos 17033

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034

This theorem is referenced by:  tlt2  29664  toslublem  29667  tosglblem  29669  isarchi2  29739  archirng  29742  archiabllem2c  29749  archiabl  29752