Theorem dalemccnedd 38496

Description: Lemma for dath 38545. 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
dalemccnedd	⊢ (𝜓 → 𝑐 ≠ 𝑑)

Proof of Theorem dalemccnedd

Step	Hyp	Ref	Expression
1		da.ps0	. . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
2		simp31 1210	. . 3 ⊢ (((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) → 𝑑 ≠ 𝑐)
3	1, 2	sylbi 216	. 2 ⊢ (𝜓 → 𝑑 ≠ 𝑐)
4	3	necomd 2997	1 ⊢ (𝜓 → 𝑐 ≠ 𝑑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107   ≠ wne 2941   class class class wbr 5147  (class class class)co 7404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-ex 1783  df-cleq 2725  df-ne 2942

This theorem is referenced by:  dalemswapyzps  38499  dalemrotps  38500  dalemcjden  38501