Metamath Proof Explorer


Theorem leat2

Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013)

Ref Expression
Hypotheses leatom.b 𝐵 = ( Base ‘ 𝐾 )
leatom.l = ( le ‘ 𝐾 )
leatom.z 0 = ( 0. ‘ 𝐾 )
leatom.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion leat2 ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ ( 𝑋0𝑋 𝑃 ) ) → 𝑋 = 𝑃 )

Proof

Step Hyp Ref Expression
1 leatom.b 𝐵 = ( Base ‘ 𝐾 )
2 leatom.l = ( le ‘ 𝐾 )
3 leatom.z 0 = ( 0. ‘ 𝐾 )
4 leatom.a 𝐴 = ( Atoms ‘ 𝐾 )
5 1 2 3 4 leatb ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) → ( 𝑋 𝑃 ↔ ( 𝑋 = 𝑃𝑋 = 0 ) ) )
6 orcom ( ( 𝑋 = 𝑃𝑋 = 0 ) ↔ ( 𝑋 = 0𝑋 = 𝑃 ) )
7 neor ( ( 𝑋 = 0𝑋 = 𝑃 ) ↔ ( 𝑋0𝑋 = 𝑃 ) )
8 6 7 bitri ( ( 𝑋 = 𝑃𝑋 = 0 ) ↔ ( 𝑋0𝑋 = 𝑃 ) )
9 5 8 bitrdi ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) → ( 𝑋 𝑃 ↔ ( 𝑋0𝑋 = 𝑃 ) ) )
10 9 biimpd ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) → ( 𝑋 𝑃 → ( 𝑋0𝑋 = 𝑃 ) ) )
11 10 com23 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) → ( 𝑋0 → ( 𝑋 𝑃𝑋 = 𝑃 ) ) )
12 11 imp32 ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ ( 𝑋0𝑋 𝑃 ) ) → 𝑋 = 𝑃 )