Theorem nulsslt 32239

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		elex 3361	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
2		0ex 4921	. . 3 ⊢ ∅ ∈ V
3	1, 2	jctil 503	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ∈ V ∧ 𝐴 ∈ V))
4		0ss 4114	. . . 4 ⊢ ∅ ⊆ No
5	4	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4305	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		ral0 4215	. . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦)
9	5, 6, 8	3jca 1121	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦))
10		brsslt 32231	. 2 ⊢ (∅ <<s 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦)))
11	3, 9, 10	sylanbrc 564	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   ∈ wcel 2144  ∀wral 3060  Vcvv 3349   ⊆ wss 3721  ∅c0 4061  𝒫 cpw 4295   class class class wbr 4784   No csur 32124   <s cslt 32125   <<s csslt 32227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-sslt 32228

This theorem is referenced by: (None)