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 ∧ 𝑃𝐴𝑄𝐴 ) → ¬ 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 ∧ 𝑃𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
5 4 3adant3 ( ( 𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
6 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 2 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
8 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
9 6 8 3 1 2 atcvreq0 ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄𝐴 ) → ( 𝑃 𝐶 𝑄𝑃 = ( 0. ‘ 𝐾 ) ) )
10 7 9 syl3an2 ( ( 𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝐶 𝑄𝑃 = ( 0. ‘ 𝐾 ) ) )
11 10 necon3bbid ( ( 𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴 ) → ( ¬ 𝑃 𝐶 𝑄𝑃 ≠ ( 0. ‘ 𝐾 ) ) )
12 5 11 mpbird ( ( 𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴 ) → ¬ 𝑃 𝐶 𝑄 )