Theorem cvrval2 34561

Description: Binary relation expressing Y covers X. Definition of covers in [Kalmbach] p. 15. (cvbr2 29142 analog.) (Contributed by NM, 16-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
cvrval2	|- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) ) )

Distinct variable groups:   z, A   z, B   z, K   z, X   z, Y

Allowed substitution hints:   C( z)   .< ( z)   .<_ ( z)

Proof of Theorem cvrval2

Step	Hyp	Ref	Expression
1		cvrletr.b	. . 3 |- B = ( Base ` K )
2		cvrletr.s	. . 3 |- .< = ( lt ` K )
3		cvrletr.c	. . 3 |- C = ( <o ` K )
4	1, 2, 3	cvrval 34556	. 2 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
5		iman 440	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
6		df-ne 2795	. . . . . . . . 9 |- ( z =/= Y <-> -. z = Y )
7	6	anbi2i 730	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
8	5, 7	xchbinxr 325	. . . . . . 7 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) )
9		cvrletr.l	. . . . . . . . . . . . 13 |- .<_ = ( le ` K )
10	9, 2	pltval 16960	. . . . . . . . . . . 12 |- ( ( K e. A /\ z e. B /\ Y e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
11	10	3com23 1271	. . . . . . . . . . 11 |- ( ( K e. A /\ Y e. B /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
12	11	3expa 1265	. . . . . . . . . 10 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
13	12	anbi2d 740	. . . . . . . . 9 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( X .< z /\ z .< Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) ) )
14		anass 681	. . . . . . . . 9 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) )
15	13, 14	syl6rbbr 279	. . . . . . . 8 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ z .< Y ) ) )
16	15	notbid 308	. . . . . . 7 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> -. ( X .< z /\ z .< Y ) ) )
17	8, 16	syl5bb 272	. . . . . 6 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( X .< z /\ z .< Y ) ) )
18	17	ralbidva 2985	. . . . 5 |- ( ( K e. A /\ Y e. B ) -> ( A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> A. z e. B -. ( X .< z /\ z .< Y ) ) )
19		ralnex 2992	. . . . 5 |- ( A. z e. B -. ( X .< z /\ z .< Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) )
20	18, 19	syl6bb 276	. . . 4 |- ( ( K e. A /\ Y e. B ) -> ( A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) ) )
21	20	anbi2d 740	. . 3 |- ( ( K e. A /\ Y e. B ) -> ( ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
22	21	3adant2 1080	. 2 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
23	4, 22	bitr4d 271	1 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ A. z e. B ( ( X .< z /\ z .<_ Y ) -> z = Y ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   <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-plt 16958  df-covers 34553

This theorem is referenced by:  isat3  34594  cvlcvr1  34626