Theorem dalemccea 39685

Description: Lemma for dath 39738. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Hypothesis

Ref	Expression
da.ps0	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))

Assertion

Ref	Expression
dalemccea	⊢ (𝜓 → 𝑐 ∈ 𝐴)

Proof of Theorem dalemccea

Step	Hyp	Ref	Expression
1		da.ps0	. 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
2		simp1l 1198	. 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝑐 ∈ 𝐴)
3	1, 2	sylbi 217	1 ⊢ (𝜓 → 𝑐 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2940   class class class wbr 5143  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089

This theorem is referenced by:  dalemcceb  39691  dalemswapyzps  39692  dalemrotps  39693  dalemcjden  39694  dalem23  39698  dalem24  39699  dalem25  39700  dalem27  39701  dalem28  39702  dalem38  39712  dalem39  39713  dalem44  39718  dalem51  39725  dalem56  39730