Theorem nulsslt 33264

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 3514	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
2		0ex 5213	. . 3 ⊢ ∅ ∈ V
3	1, 2	jctil 522	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ∈ V ∧ 𝐴 ∈ V))
4		0ss 4352	. . . 4 ⊢ ∅ ⊆ No
5	4	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi 4550	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		ral0 4458	. . . 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 1124	. 2 ⊢ (𝐴 ∈ 𝒫 No → (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦))
10		brsslt 33256	. 2 ⊢ (∅ <<s 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥 <s 𝑦)))
11	3, 9, 10	sylanbrc 585	1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541   class class class wbr 5068   No csur 33149   <s cslt 33150   <<s csslt 33252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-sslt 33253

This theorem is referenced by: (None)