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Metamath Proof Explorer

Theorem cvrnrefN

Description: The covers relation is not reflexive. ( cvnref analog.) (Contributed by NM, 1-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cvrne.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvrne.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion cvrnrefN ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 𝐶 𝑋 )

Proof

Step Hyp Ref Expression
1 cvrne.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrne.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3 eqid ⊢ 𝑋 = 𝑋
4 simpll ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝐾 ∈ 𝐴 )
5 simplr ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ∈ 𝐵 )
6 simpr ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 𝐶 𝑋 )
7 1 2 cvrne ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ≠ 𝑋 )
8 4 5 5 6 7 syl31anc ⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ≠ 𝑋 )
9 8 ex ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑋 → 𝑋 ≠ 𝑋 ) )
10 9 necon2bd ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑋 → ¬ 𝑋 𝐶 𝑋 ) )
11 3 10 mpi ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 𝐶 𝑋 )