Theorem posasymb 16776

Description: A poset ordering is asymetric.

Hypotheses

Ref	Expression
poslem.b	|- B = (base` K)
poslem.l	|- L = (le` K)

Assertion

Ref	Expression
posasymb	|- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XLY /\ YLX) <-> X = Y))

Proof of Theorem posasymb

Step	Hyp	Ref	Expression
1		simp1 876	. . . 4 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> K e. PosetNEW)
2		simp2 877	. . . 4 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> X e. B)
3		simp3 878	. . . 4 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> Y e. B)
4		poslem.b	. . . . 5 |- B = (base` K)
5		poslem.l	. . . . 5 |- L = (le` K)
6	4, 5	poslem 16774	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Y e. B)) -> (XLX /\ ((XLY /\ YLX) -> X = Y) /\ ((XLY /\ YLY) -> XLY)))
7	1, 2, 3, 3, 6	syl13anc 1102	. . 3 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (XLX /\ ((XLY /\ YLX) -> X = Y) /\ ((XLY /\ YLY) -> XLY)))
8		simp2 877	. . 3 |- ((XLX /\ ((XLY /\ YLX) -> X = Y) /\ ((XLY /\ YLY) -> XLY)) -> ((XLY /\ YLX) -> X = Y))
9	7, 8	syl 12	. 2 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XLY /\ YLX) -> X = Y))
10		breq2 3342	. . . . 5 |- (X = Y -> (XLX <-> XLY))
11	4, 5	posref 16775	. . . . 5 |- ((K e. PosetNEW /\ X e. B) -> XLX)
12	10, 11	syl5cbi 226	. . . 4 |- ((K e. PosetNEW /\ X e. B) -> (X = Y -> XLY))
13		breq1 3341	. . . . 5 |- (X = Y -> (XLX <-> YLX))
14	13, 11	syl5cbi 226	. . . 4 |- ((K e. PosetNEW /\ X e. B) -> (X = Y -> YLX))
15	12, 14	jcad 661	. . 3 |- ((K e. PosetNEW /\ X e. B) -> (X = Y -> (XLY /\ YLX)))
16	15	3adant3 896	. 2 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (X = Y -> (XLY /\ YLX)))
17	9, 16	impbid 574	1 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XLY /\ YLX) <-> X = Y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  basecbs 16758  lecple 16759  PosetNEWcpo 16760

This theorem is referenced by:  pltnle 16786  pltval3 16787  posgelem 16795  lubid 16807  latasymb 16856  latleeqj1 16864  latleeqm1 16874  latjass 16886  ople0 16917  op1le 16919  oldmm1 16946

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-rep 3428  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-3an 860  df-tru 1262  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-struct 16708  df-poset 16772

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