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Theorem cvrnrefN 39187
Description: The covers relation is not reflexive. (cvnref 32314 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrne.b 𝐵 = (Base‘𝐾)
cvrne.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnrefN ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)

Proof of Theorem cvrnrefN
StepHypRef Expression
1 eqid 2734 . 2 𝑋 = 𝑋
2 simpll 766 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝐾𝐴)
3 simplr 768 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐵)
4 simpr 484 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐶𝑋)
5 cvrne.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrne.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 39186 . . . . 5 (((𝐾𝐴𝑋𝐵𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
82, 3, 3, 4, 7syl31anc 1373 . . . 4 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
98ex 412 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋𝐶𝑋𝑋𝑋))
109necon2bd 2958 . 2 ((𝐾𝐴𝑋𝐵) → (𝑋 = 𝑋 → ¬ 𝑋𝐶𝑋))
111, 10mpi 20 1 ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2103  wne 2942   class class class wbr 5169  cfv 6572  Basecbs 17253  ccvr 39167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-plt 18395  df-covers 39171
This theorem is referenced by: (None)
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