Theorem sltssepc 28001

Description: Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
sltssepc	⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)

Proof of Theorem sltssepc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sltssep 27997	. . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
2		breq1 5117	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑦 ↔ 𝑋 <s 𝑦))
3		breq2 5118	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 <s 𝑦 ↔ 𝑋 <s 𝑌))
4	2, 3	rspc2va 3596	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌)
5	4	ancoms 464	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌)
6	1, 5	sylan 592	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌)
7	6	3impb 1132	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146  ∀wral 3082   class class class wbr 5114   <s clts 27842   <<s cslts 27987

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-slts 27988

This theorem is used by:  sltssepcd  28002  sltsun1  28018  sltsun2  28019  sltsdisj  28033  cofcutr  28154