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Metamath Proof Explorer

Theorem atncvrN

Description: Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atncvr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
atncvr.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atncvrN ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 𝐶 𝑄 )

Proof

Step Hyp Ref Expression
1 atncvr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
2 atncvr.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
4 3 2 atn0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
5 4 3adant3 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 2 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
8 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
9 6 8 3 1 2 atcvreq0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 𝐶 𝑄 ↔ 𝑃 = ( 0. ‘ 𝐾 ) ) )
10 7 9 syl3an2 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 𝐶 𝑄 ↔ 𝑃 = ( 0. ‘ 𝐾 ) ) )
11 10 necon3bbid ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑃 𝐶 𝑄 ↔ 𝑃 ≠ ( 0. ‘ 𝐾 ) ) )
12 5 11 mpbird ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 𝐶 𝑄 )