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Theorem nulsslt 33979
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 5235 . . 3 ∅ ∈ V
21a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3 id 22 . 2 (𝐴 ∈ 𝒫 No 𝐴 ∈ 𝒫 No )
4 0ss 4336 . . 3 ∅ ⊆ No
54a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6 elpwi 4548 . 2 (𝐴 ∈ 𝒫 No 𝐴 No )
7 noel 4270 . . . 4 ¬ 𝑥 ∈ ∅
87pm2.21i 119 . . 3 (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
983ad2ant2 1133 . 2 ((𝐴 ∈ 𝒫 No 𝑥 ∈ ∅ ∧ 𝑦𝐴) → 𝑥 <s 𝑦)
102, 3, 5, 6, 9ssltd 33974 1 (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3431  wss 3892  c0 4262  𝒫 cpw 4539   class class class wbr 5079   No csur 33831   <s cslt 33832   <<s csslt 33963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5595  df-sslt 33964
This theorem is referenced by:  bday0s  34010  bday0b  34012
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