Theorem ssltsepcd 33643

Description: Two elements of separated sets obey less than. Deduction form of ssltsepc 33642. (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 33642	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
5	1, 2, 3, 4	syl3anc 1372	1 ⊢ (𝜑 → 𝑋 <s 𝑌)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5040   <s cslt 33499   <<s csslt 33630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-xp 5541  df-sslt 33631

This theorem is referenced by:  sslttr  33656  cofsslt  33743  coinitsslt  33744  cofcutrtime  33748