Theorem nulsltsd 27794

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 4542	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 No )
4		nulslts 27792	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
5	3, 4	syl 17	1 ⊢ (𝜑 → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2119   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536   class class class wbr 5079   No csur 27628   <<s cslts 27774

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-slts 27775

This theorem is referenced by: (None)