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Theorem cvrnrefN 37676
Description: The covers relation is not reflexive. (cvnref 31062 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 2738 . 2 𝑋 = 𝑋
2 simpll 766 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝐾𝐴)
3 simplr 768 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐵)
4 simpr 486 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐶𝑋)
5 cvrne.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrne.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 37675 . . . . 5 (((𝐾𝐴𝑋𝐵𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
82, 3, 3, 4, 7syl31anc 1374 . . . 4 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
98ex 414 . . 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 397   = wceq 1542  wcel 2107  wne 2942   class class class wbr 5104  cfv 6494  Basecbs 17043  ccvr 37656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-plt 18179  df-covers 37660
This theorem is referenced by: (None)
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