Theorem lhpexle2

Description: There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013)

	Ref	Expression
Hypotheses	lhpex1.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpex1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpex1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpexle2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		lhpex1.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhpex1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lhpex1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4	1 2 3	lhpexle1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ) )
5	1 2 3	lhpexle1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ) )
6	5	adantr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ) )
7	1 2 3	lhpexle2lem	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋 ) )
8	7	3expa	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋 ) )
9	6 8	lhpexle1lem	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋 ) )
10		3ancomb	⊢ ( ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋 ) ↔ ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ) )
11	10	rexbii	⊢ ( ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑌 ∧ 𝑝 ≠ 𝑋 ) ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ) )
12	9 11	sylib	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ) )
13	4 12	lhpexle1lem	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑊 ∧ 𝑝 ≠ 𝑋 ∧ 𝑝 ≠ 𝑌 ) )