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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 ≠ 0 ∧ 𝑋 ≤ 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ) )
6		orcom	⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) )
7		neor	⊢ ( ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) )
8	6 7	bitri	⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) )
9	5 8	bitrdi	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) )
10	9	biimpd	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 → ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) )
11	10	com23	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≠ 0 → ( 𝑋 ≤ 𝑃 → 𝑋 = 𝑃 ) ) )
12	11	imp32	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃 ) ) → 𝑋 = 𝑃 )