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Theorem nulssgt 32034
Description: The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
nulssgt (𝐴 ∈ 𝒫 No 𝐴 <<s ∅)

Proof of Theorem nulssgt
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3243 . . 3 (𝐴 ∈ 𝒫 No 𝐴 ∈ V)
2 0ex 4823 . . 3 ∅ ∈ V
31, 2jctir 560 . 2 (𝐴 ∈ 𝒫 No → (𝐴 ∈ V ∧ ∅ ∈ V))
4 elpwi 4201 . . 3 (𝐴 ∈ 𝒫 No 𝐴 No )
5 0ss 4005 . . . 4 ∅ ⊆ No
65a1i 11 . . 3 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7 ral0 4109 . . . . 5 𝑦 ∈ ∅ 𝑥 <s 𝑦
87rgenw 2953 . . . 4 𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦
98a1i 11 . . 3 (𝐴 ∈ 𝒫 No → ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦)
104, 6, 93jca 1261 . 2 (𝐴 ∈ 𝒫 No → (𝐴 No ∧ ∅ ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦))
11 brsslt 32025 . 2 (𝐴 <<s ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 No ∧ ∅ ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦)))
123, 10, 11sylanbrc 699 1 (𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685   No csur 31918   <s cslt 31919   <<s csslt 32021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-sslt 32022
This theorem is referenced by: (None)
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