Theorem nulsslt 27766

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

Assertion

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

Proof of Theorem nulsslt

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

Step	Hyp	Ref	Expression
1		0ex 5282	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		id 22	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
4		0ss 4380	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4587	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel 4318	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i 119	. . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant2 1134	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦)
10	2, 3, 5, 6, 9	ssltd 27760	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580   class class class wbr 5124   No csur 27608   <s cslt 27609   <<s csslt 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-sslt 27750

This theorem is referenced by:  bday0s  27797  bday0b  27799