Theorem nulsltsd 27928

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 4564	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 No )
4		nulslts 27926	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
5	3, 4	syl 18	1 ⊢ (𝜑 → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2145   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105   No csur 27762   <<s cslts 27908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-slts 27909

This theorem is referenced by: (None)