Theorem nulslts 27781

Description: The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
nulslts	⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Proof of Theorem nulslts

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

Step	Hyp	Ref	Expression
1		0ex 5242	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		elex 3451	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
4		0ss 4341	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4549	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel 4279	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i 119	. . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦)
10	2, 3, 5, 6, 9	sltsd 27774	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542   class class class wbr 5086   No csur 27617   <s clts 27618   <<s cslts 27763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-slts 27764

This theorem is referenced by:  nulsltsd  27783  bday0  27817  bday0b  27819