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Theorem nulsslt 27158
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.
StepHypRef Expression
1 0ex 5265 . . 3 ∅ ∈ V
21a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3 id 22 . 2 (𝐴 ∈ 𝒫 No 𝐴 ∈ 𝒫 No )
4 0ss 4357 . . 3 ∅ ⊆ No
54a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6 elpwi 4568 . 2 (𝐴 ∈ 𝒫 No 𝐴 No )
7 noel 4291 . . . 4 ¬ 𝑥 ∈ ∅
87pm2.21i 119 . . 3 (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
983ad2ant2 1135 . 2 ((𝐴 ∈ 𝒫 No 𝑥 ∈ ∅ ∧ 𝑦𝐴) → 𝑥 <s 𝑦)
102, 3, 5, 6, 9ssltd 27153 1 (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444  wss 3911  c0 4283  𝒫 cpw 4561   class class class wbr 5106   No csur 27004   <s cslt 27005   <<s csslt 27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-sslt 27143
This theorem is referenced by:  bday0s  27189  bday0b  27191
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