Theorem dalemclccjdd 40260

Description: Lemma for dath 40308. 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
dalemclccjdd	⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))

Proof of Theorem dalemclccjdd

Step	Hyp	Ref	Expression
1		da.ps0	. 2 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
2		simp33 1221	. 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝐶 ≤ (𝑐 ∨ 𝑑))
3	1, 2	sylbi 219	1 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   ∈ wcel 2136   ≠ wne 2951   class class class wbr 5094  (class class class)co 7385

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 209  df-an 399  df-3an 1097

This theorem is referenced by:  dalemswapyzps  40262  dalemrotps  40263  dalem21  40266  dalem25  40270