Theorem posasymb 16952

Description: A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.)

Hypotheses

Ref	Expression
posi.b	|- B = ( Base ` K )
posi.l	|- .<_ = ( le ` K )

Assertion

Ref	Expression
posasymb	|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )

Proof of Theorem posasymb

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> K e. Poset )
2		simp2 1062	. . . 4 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> X e. B )
3		simp3 1063	. . . 4 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> Y e. B )
4		posi.b	. . . . 5 |- B = ( Base ` K )
5		posi.l	. . . . 5 |- .<_ = ( le ` K )
6	4, 5	posi 16950	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Y e. B ) ) -> ( X .<_ X /\ ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) /\ ( ( X .<_ Y /\ Y .<_ Y ) -> X .<_ Y ) ) )
7	1, 2, 3, 3, 6	syl13anc 1328	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ X /\ ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) /\ ( ( X .<_ Y /\ Y .<_ Y ) -> X .<_ Y ) ) )
8	7	simp2d 1074	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) )
9	4, 5	posref 16951	. . . . 5 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )
10		breq2 4657	. . . . 5 |- ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
11	9, 10	syl5ibcom 235	. . . 4 |- ( ( K e. Poset /\ X e. B ) -> ( X = Y -> X .<_ Y ) )
12		breq1 4656	. . . . 5 |- ( X = Y -> ( X .<_ X <-> Y .<_ X ) )
13	9, 12	syl5ibcom 235	. . . 4 |- ( ( K e. Poset /\ X e. B ) -> ( X = Y -> Y .<_ X ) )
14	11, 13	jcad 555	. . 3 |- ( ( K e. Poset /\ X e. B ) -> ( X = Y -> ( X .<_ Y /\ Y .<_ X ) ) )
15	14	3adant3 1081	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X = Y -> ( X .<_ Y /\ Y .<_ X ) ) )
16	8, 15	impbid 202	1 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )

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

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-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-iota 5851  df-fv 5896  df-preset 16928  df-poset 16946

This theorem is referenced by:  pltnle  16966  pltval3  16967  lublecllem  16988  latasymb  17054  latleeqj1  17063  latleeqm1  17079  odupos  17135  poslubmo  17146  posglbmo  17147  posrasymb  29657  archirngz  29743  archiabllem1a  29745  ople0  34474  op1le  34479  atlle0  34592