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Theorem cvrnrefN 37790
Description: The covers relation is not reflexive. (cvnref 31275 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 2733 . 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 37789 . . . . 5 (((𝐾𝐴𝑋𝐵𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
82, 3, 3, 4, 7syl31anc 1374 . . . 4 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
98ex 414 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋𝐶𝑋𝑋𝑋))
109necon2bd 2956 . 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 2940   class class class wbr 5106  cfv 6497  Basecbs 17088  ccvr 37770
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 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-plt 18224  df-covers 37774
This theorem is referenced by: (None)
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