Theorem nulslts 28194

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 5265	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		elex 3475	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
4		0ss 4353	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No)
6		elpwi 4567	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No)
7		noel 4286	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦)
10	2, 3, 5, 6, 9	sltsd 28187	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106  Nocsur 28030   <s clts 28031   <<s cslts 28176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-slts 28177

This theorem is used by:  nulsltsd  28196  bday0  28230  bday0b  28232