Theorem nulslts 27926

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 5262	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3		elex 3478	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
4		0ss 4357	. . 3 ⊢ ∅ ⊆ No
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4565	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel 4293	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant2 1150	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦)
10	2, 3, 5, 6, 9	sltsd 27919	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105   No csur 27762   <s clts 27763   <<s cslts 27908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-slts 27909

This theorem is referenced by:  nulsltsd  27928  bday0  27962  bday0b  27964