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Theorem cvrnrefN 38794
Description: The covers relation is not reflexive. (cvnref 32129 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 2728 . 2 𝑋 = 𝑋
2 simpll 765 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝐾𝐴)
3 simplr 767 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐵)
4 simpr 483 . . . . 5 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝐶𝑋)
5 cvrne.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrne.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 38793 . . . . 5 (((𝐾𝐴𝑋𝐵𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
82, 3, 3, 4, 7syl31anc 1370 . . . 4 (((𝐾𝐴𝑋𝐵) ∧ 𝑋𝐶𝑋) → 𝑋𝑋)
98ex 411 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋𝐶𝑋𝑋𝑋))
109necon2bd 2953 . 2 ((𝐾𝐴𝑋𝐵) → (𝑋 = 𝑋 → ¬ 𝑋𝐶𝑋))
111, 10mpi 20 1 ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937   class class class wbr 5152  cfv 6553  Basecbs 17189  ccvr 38774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-plt 18331  df-covers 38778
This theorem is referenced by: (None)
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