Theorem ltltncvr 33386

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 1030	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X e. B )
3		simplr3 1032	. . . 4 |- ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Z e. B )
4		simplr2 1031	. . . 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 33235	. . . 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 1232	. . 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 965   = wceq 1370   e. wcel 1758   class class class wbr 4395   ` cfv 5521   Basecbs 14287   ltcplt 15225   <o ccvr 33226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-covers 33230

This theorem is referenced by:  ltcvrntr  33387