Theorem dalemccea 39212

Description: Lemma for dath 39265. 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 1194	. 2 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝑐 ∈ 𝐴)
3	1, 2	sylbi 216	1 ⊢ (𝜓 → 𝑐 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098   ≠ wne 2930   class class class wbr 5143  (class class class)co 7416

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 206  df-an 395  df-3an 1086

This theorem is referenced by:  dalemcceb  39218  dalemswapyzps  39219  dalemrotps  39220  dalemcjden  39221  dalem23  39225  dalem24  39226  dalem25  39227  dalem27  39228  dalem28  39229  dalem38  39239  dalem39  39240  dalem44  39245  dalem51  39252  dalem56  39257