Theorem cvrletrN 34560

Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem cvrletrN

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> K e. Poset )
2		simplr1 1103	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X e. B )
3		simplr2 1104	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> Y e. B )
4		simpr 477	. . . 4 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X C Y )
5		cvrletr.b	. . . . 5 |- B = ( Base ` K )
6		cvrletr.s	. . . . 5 |- .< = ( lt ` K )
7		cvrletr.c	. . . . 5 |- C = ( <o ` K )
8	5, 6, 7	cvrlt 34557	. . . 4 |- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .< Y )
9	1, 2, 3, 4, 8	syl31anc 1329	. . 3 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> X .< Y )
10		cvrletr.l	. . . . 5 |- .<_ = ( le ` K )
11	5, 10, 6	pltletr 16971	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .<_ Z ) -> X .< Z ) )
12	11	adantr 481	. . 3 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> ( ( X .< Y /\ Y .<_ Z ) -> X .< Z ) )
13	9, 12	mpand 711	. 2 |- ( ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Y ) -> ( Y .<_ Z -> X .< Z ) )
14	13	expimpd 629	1 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ Y .<_ Z ) -> X .< Z ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   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-preset 16928  df-poset 16946  df-plt 16958  df-covers 34553

This theorem is referenced by: (None)