Theorem nulsslt 33991

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 5231	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		id 22	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
4		0ss 4330	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4542	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel 4264	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i 119	. . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant2 1133	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦)
10	2, 3, 5, 6, 9	ssltd 33986	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533   class class class wbr 5074   No csur 33843   <s cslt 33844   <<s csslt 33975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-sslt 33976

This theorem is referenced by:  bday0s  34022  bday0b  34024