Theorem dalemcceb

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012)

	Ref	Expression
Hypotheses	da.ps0	⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
	da.a1	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	dalemcceb	⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		da.ps0	⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
2		da.a1	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3	1	dalemccea	⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5	4 2	atbase	⊢ ( 𝑐 ∈ 𝐴 → 𝑐 ∈ ( Base ‘ 𝐾 ) )
6	3 5	syl	⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) )