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Theorem cvrnrefN 29472
Description: The covers relation is not reflexive. (cvnref 22871 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrne.b  |-  B  =  ( Base `  K
)
cvrne.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnrefN  |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X C X )

Proof of Theorem cvrnrefN
StepHypRef Expression
1 eqid 2283 . 2  |-  X  =  X
2 simpll 730 . . . . 5  |-  ( ( ( K  e.  A  /\  X  e.  B
)  /\  X C X )  ->  K  e.  A )
3 simplr 731 . . . . 5  |-  ( ( ( K  e.  A  /\  X  e.  B
)  /\  X C X )  ->  X  e.  B )
4 simpr 447 . . . . 5  |-  ( ( ( K  e.  A  /\  X  e.  B
)  /\  X C X )  ->  X C X )
5 cvrne.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cvrne.c . . . . . 6  |-  C  =  (  <o  `  K )
75, 6cvrne 29471 . . . . 5  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  X  e.  B
)  /\  X C X )  ->  X  =/=  X )
82, 3, 3, 4, 7syl31anc 1185 . . . 4  |-  ( ( ( K  e.  A  /\  X  e.  B
)  /\  X C X )  ->  X  =/=  X )
98ex 423 . . 3  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X C X  ->  X  =/=  X
) )
109necon2bd 2495 . 2  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  =  X  ->  -.  X C X ) )
111, 10mpi 16 1  |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X C X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-plt 14092  df-covers 29456
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