Theorem leat3

Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012)

	Ref	Expression
Hypotheses	leatom.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	leatom.l	⊢ ≤ = ( le ‘ 𝐾 )
	leatom.z	⊢ 0 = ( 0. ‘ 𝐾 )
	leatom.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	leat3	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ) )

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	leat	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) )
6		simpl3	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → 𝑃 ∈ 𝐴 )
7		eleq1a	⊢ ( 𝑃 ∈ 𝐴 → ( 𝑋 = 𝑃 → 𝑋 ∈ 𝐴 ) )
8	6 7	syl	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 = 𝑃 → 𝑋 ∈ 𝐴 ) )
9	8	orim1d	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) → ( 𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ) ) )
10	5 9	mpd	⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ) )