Theorem nulsslt 31908

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 3212	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
2		0ex 4790	. . 3 ⊢ ∅ ∈ V
3	1, 2	jctil 560	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ∈ V ∧ 𝐴 ∈ V))
4		0ss 3972	. . . 4 ⊢ ∅ ⊆ No
5	4	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4168	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		ral0 4076	. . . 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 1242	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦))
10		brsslt 31900	. 2 ⊢ (∅ <<s 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦)))
11	3, 9, 10	sylanbrc 698	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653   No csur 31793   <s cslt 31794   <<s csslt 31896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-sslt 31897

This theorem is referenced by: (None)