Theorem nulsltsd 28050

Description: The empty set is less-than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)

Hypotheses

Ref	Expression
nulsltsd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
nulsltsd.2	⊢ (𝜑 → 𝐴 ⊆ No )

Assertion

Ref	Expression
nulsltsd	⊢ (𝜑 → ∅ <<s 𝐴)

Proof of Theorem nulsltsd

Step	Hyp	Ref	Expression
1		nulsltsd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		nulsltsd.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ No )
3	1, 2	elpwd 4566	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 No )
4		nulslts 28048	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
5	3, 4	syl 18	1 ⊢ (𝜑 → ∅ <<s 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5107   No csur 27884   <<s cslts 28030

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 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-slts 28031

This theorem is used by: (None)