Theorem cvrcmp 34570

Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)

Hypotheses

Ref	Expression
cvrcmp.b	|- B = ( Base ` K )
cvrcmp.l	|- .<_ = ( le ` K )
cvrcmp.c	|- C = ( <o ` K )

Assertion

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

Proof of Theorem cvrcmp

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . 5 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> K e. Poset )
2		simpl23 1141	. . . . 5 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z e. B )
3		simpl21 1139	. . . . 5 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X e. B )
4		simpl3l 1116	. . . . 5 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z C X )
5		cvrcmp.b	. . . . . 6 |- B = ( Base ` K )
6		cvrcmp.c	. . . . . 6 |- C = ( <o ` K )
7	5, 6	cvrne 34568	. . . . 5 |- ( ( ( K e. Poset /\ Z e. B /\ X e. B ) /\ Z C X ) -> Z =/= X )
8	1, 2, 3, 4, 7	syl31anc 1329	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z =/= X )
9		cvrcmp.l	. . . . . . . 8 |- .<_ = ( le ` K )
10	5, 9, 6	cvrle 34565	. . . . . . 7 |- ( ( ( K e. Poset /\ Z e. B /\ X e. B ) /\ Z C X ) -> Z .<_ X )
11	1, 2, 3, 4, 10	syl31anc 1329	. . . . . 6 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z .<_ X )
12		simpr 477	. . . . . 6 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X .<_ Y )
13		simpl22 1140	. . . . . . 7 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Y e. B )
14		simpl3r 1117	. . . . . . 7 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> Z C Y )
15	5, 9, 6	cvrnbtwn4 34566	. . . . . . 7 |- ( ( K e. Poset /\ ( Z e. B /\ Y e. B /\ X e. B ) /\ Z C Y ) -> ( ( Z .<_ X /\ X .<_ Y ) <-> ( Z = X \/ X = Y ) ) )
16	1, 2, 13, 3, 14, 15	syl131anc 1339	. . . . . 6 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( ( Z .<_ X /\ X .<_ Y ) <-> ( Z = X \/ X = Y ) ) )
17	11, 12, 16	mpbi2and 956	. . . . 5 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( Z = X \/ X = Y ) )
18		neor 2885	. . . . 5 |- ( ( Z = X \/ X = Y ) <-> ( Z =/= X -> X = Y ) )
19	17, 18	sylib 208	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> ( Z =/= X -> X = Y ) )
20	8, 19	mpd 15	. . 3 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) /\ X .<_ Y ) -> X = Y )
21	20	ex 450	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X .<_ Y -> X = Y ) )
22		simp1 1061	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> K e. Poset )
23		simp21 1094	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> X e. B )
24	5, 9	posref 16951	. . . 4 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )
25	22, 23, 24	syl2anc 693	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> X .<_ X )
26		breq2 4657	. . 3 |- ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
27	25, 26	syl5ibcom 235	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X = Y -> X .<_ Y ) )
28	21, 27	impbid 202	1 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( Z C X /\ Z C Y ) ) -> ( X .<_ Y <-> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   <o ccvr 34549

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-covers 34553

This theorem is referenced by:  cvrcmp2  34571  atcmp  34598  llncmp  34808  lplncmp  34848  lvolcmp  34903  lhp2lt  35287