Theorem cdleme9taN

Description: Part of proof of Lemma E in Crawley p. 113. X represents t_1, which we prove is an atom. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme8t.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme8t.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme8t.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme8t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme8t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme8t.x	⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
Assertion	cdleme9taN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑇 ) ) → 𝑋 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cdleme8t.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme8t.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme8t.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme8t.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme8t.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme8t.x	⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
7	1 2 3 4 5 6	cdleme9a	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑇 ) ) → 𝑋 ∈ 𝐴 )