Theorem nulsslt 27860

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 5325	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		id 22	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
4		0ss 4423	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4629	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel 4360	. . . 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 27854	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622   class class class wbr 5166   No csur 27702   <s cslt 27703   <<s csslt 27843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-sslt 27844

This theorem is referenced by:  bday0s  27891  bday0b  27893