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Theorem nulsslt 27748
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 5309 . . 3 ∅ ∈ V
21a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ∈ V)
3 id 22 . 2 (𝐴 ∈ 𝒫 No 𝐴 ∈ 𝒫 No )
4 0ss 4398 . . 3 ∅ ⊆ No
54a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6 elpwi 4611 . 2 (𝐴 ∈ 𝒫 No 𝐴 No )
7 noel 4332 . . . 4 ¬ 𝑥 ∈ ∅
87pm2.21i 119 . . 3 (𝑥 ∈ ∅ → 𝑥 <s 𝑦)
983ad2ant2 1131 . 2 ((𝐴 ∈ 𝒫 No 𝑥 ∈ ∅ ∧ 𝑦𝐴) → 𝑥 <s 𝑦)
102, 3, 5, 6, 9ssltd 27742 1 (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3471  wss 3947  c0 4324  𝒫 cpw 4604   class class class wbr 5150   No csur 27591   <s cslt 27592   <<s csslt 27731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-sslt 27732
This theorem is referenced by:  bday0s  27779  bday0b  27781
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