Theorem ssltsepcd 33988

Description: Two elements of separated sets obey less than. Deduction form of ssltsepc 33987. (Contributed by Scott Fenton, 25-Sep-2024.)

Hypotheses

Ref	Expression
ssltsepcd.1	⊢ (𝜑 → 𝐴 <<s 𝐵)
ssltsepcd.2	⊢ (𝜑 → 𝑋 ∈ 𝐴)
ssltsepcd.3	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
ssltsepcd	⊢ (𝜑 → 𝑋 <s 𝑌)

Proof of Theorem ssltsepcd

Step	Hyp	Ref	Expression
1		ssltsepcd.1	. 2 ⊢ (𝜑 → 𝐴 <<s 𝐵)
2		ssltsepcd.2	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
3		ssltsepcd.3	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ssltsepc 33987	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
5	1, 2, 3, 4	syl3anc 1370	1 ⊢ (𝜑 → 𝑋 <s 𝑌)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5074   <s cslt 33844   <<s csslt 33975

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-sslt 33976

This theorem is referenced by:  sslttr  34001  cofsslt  34088  coinitsslt  34089  cofcutrtime  34093