Theorem nulsltsd 27773

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 4560	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 No )
4		nulslts 27771	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
5	3, 4	syl 17	1 ⊢ (𝜑 → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5098   No csur 27607   <<s cslts 27753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-slts 27754

This theorem is referenced by: (None)