Theorem ltltncvr 35290

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 753	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> K e. A )
2		simplr1 1038	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X e. B )
3		simplr3 1040	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Z e. B )
4		simplr2 1039	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Y e. B )
5		simpr 461	. . . 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 35139	. . . 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 1241	. . 3 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> -. ( X .< Y /\ Y .< Z ) )
11	10	ex 434	. 2 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Z -> -. ( X .< Y /\ Y .< Z ) ) )
12	11	con2d 115	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 369   /\ w3a 973   = wceq 1395   e. wcel 1819   class class class wbr 4456   ` cfv 5594   Basecbs 14644   ltcplt 15697   <o ccvr 35130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-covers 35134

This theorem is referenced by:  ltcvrntr  35291