Theorem ltltncvr 29905

Description: A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)

Hypotheses

Ref	Expression
ltltncvr.b	|- B = ( Base ` K )
ltltncvr.s	|- .< = ( lt ` K )
ltltncvr.c	|- C = ( <o ` K )

Assertion

Ref	Expression
ltltncvr	|- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> -. X C Z ) )

Proof of Theorem ltltncvr

Step	Hyp	Ref	Expression
1		simpll 731	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> K e. A )
2		simplr1 999	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X e. B )
3		simplr3 1001	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Z e. B )
4		simplr2 1000	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Y e. B )
5		simpr 448	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X C Z )
6		ltltncvr.b	. . . . 5 |- B = ( Base ` K )
7		ltltncvr.s	. . . . 5 |- .< = ( lt ` K )
8		ltltncvr.c	. . . . 5 |- C = ( <o ` K )
9	6, 7, 8	cvrnbtwn 29754	. . . 4 |- ( ( K e. A /\ ( X e. B /\ Z e. B /\ Y e. B ) /\ X C Z ) -> -. ( X .< Y /\ Y .< Z ) )
10	1, 2, 3, 4, 5, 9	syl131anc 1197	. . 3 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> -. ( X .< Y /\ Y .< Z ) )
11	10	ex 424	. 2 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Z -> -. ( X .< Y /\ Y .< Z ) ) )
12	11	con2d 109	1 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> -. X C Z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   ltcplt 14353   <o ccvr 29745

This theorem is referenced by:  ltcvrntr  29906

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-covers 29749