Theorem dalem-ccly 34971

Description: Lemma for dath 35022. 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
dalem-ccly	⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)

Proof of Theorem dalem-ccly

Step	Hyp	Ref	Expression
1		da.ps0	. 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
2	1	simp2bi 1077	1 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ≠ wne 2794   class class class wbr 4653  (class class class)co 6650

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  dalemswapyzps  34976  dalemrotps  34977  dalem21  34980  dalem23  34982  dalem24  34983  dalem39  34997  dalem48  35006