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Theorem nulssgt 33371
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 3462 . . 3 (𝐴 ∈ 𝒫 No 𝐴 ∈ V)
2 0ex 5178 . . 3 ∅ ∈ V
31, 2jctir 524 . 2 (𝐴 ∈ 𝒫 No → (𝐴 ∈ V ∧ ∅ ∈ V))
4 elpwi 4509 . . 3 (𝐴 ∈ 𝒫 No 𝐴 No )
5 0ss 4307 . . . 4 ∅ ⊆ No
65a1i 11 . . 3 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7 ral0 4417 . . . . 5 𝑦 ∈ ∅ 𝑥 <s 𝑦
87rgenw 3121 . . . 4 𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦
98a1i 11 . . 3 (𝐴 ∈ 𝒫 No → ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦)
104, 6, 93jca 1125 . 2 (𝐴 ∈ 𝒫 No → (𝐴 No ∧ ∅ ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦))
11 brsslt 33362 . 2 (𝐴 <<s ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 No ∧ ∅ ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥 <s 𝑦)))
123, 10, 11sylanbrc 586 1 (𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2112  wral 3109  Vcvv 3444  wss 3884  c0 4246  𝒫 cpw 4500   class class class wbr 5033   No csur 33255   <s cslt 33256   <<s csslt 33358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-sslt 33359
This theorem is referenced by: (None)
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