Theorem cvrval2 29757

Description: Binary relation expressing Y covers X. Definition of covers in [Kalmbach] p. 15. (cvbr2 23739 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 29752	. 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 414	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
6		df-ne 2569	. . . . . . . . 9 |- ( z =/= Y <-> -. z = Y )
7	6	anbi2i 676	. . . . . . . 8 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( ( X .< z /\ z .<_ Y ) /\ -. z = Y ) )
8	5, 7	xchbinxr 303	. . . . . . 7 |- ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) )
9		cvrletr.l	. . . . . . . . . . . . 13 |- .<_ = ( le ` K )
10	9, 2	pltval 14372	. . . . . . . . . . . 12 |- ( ( K e. A /\ z e. B /\ Y e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
11	10	3com23 1159	. . . . . . . . . . 11 |- ( ( K e. A /\ Y e. B /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
12	11	3expa 1153	. . . . . . . . . 10 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( z .< Y <-> ( z .<_ Y /\ z =/= Y ) ) )
13	12	anbi2d 685	. . . . . . . . 9 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( X .< z /\ z .< Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) ) )
14		anass 631	. . . . . . . . 9 |- ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ ( z .<_ Y /\ z =/= Y ) ) )
15	13, 14	syl6rbbr 256	. . . . . . . 8 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> ( X .< z /\ z .< Y ) ) )
16	15	notbid 286	. . . . . . 7 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( -. ( ( X .< z /\ z .<_ Y ) /\ z =/= Y ) <-> -. ( X .< z /\ z .< Y ) ) )
17	8, 16	syl5bb 249	. . . . . 6 |- ( ( ( K e. A /\ Y e. B ) /\ z e. B ) -> ( ( ( X .< z /\ z .<_ Y ) -> z = Y ) <-> -. ( X .< z /\ z .< Y ) ) )
18	17	ralbidva 2682	. . . . 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 2676	. . . . 5 |- ( A. z e. B -. ( X .< z /\ z .< Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) )
20	18, 19	syl6bb 253	. . . 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 685	. . 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 976	. 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 248	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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   ltcplt 14353   <o ccvr 29745

This theorem is referenced by:  isat3  29790  cvlcvr1  29822

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-plt 14370  df-covers 29749