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Theorem nla0001 42012
Description: Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
Hypothesis
Ref Expression
nla0001.defsslt < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
Assertion
Ref Expression
nla0001 (𝜑 → ∅ < ∅)
Distinct variable groups:   𝑎,𝑏,𝑥,𝑦   𝑅,𝑎,𝑏   𝑆,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   < (𝑥,𝑦,𝑎,𝑏)

Proof of Theorem nla0001
StepHypRef Expression
1 nla0001.defsslt . 2 < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
2 0ex 5301 . . 3 ∅ ∈ V
32a1i 11 . 2 (𝜑 → ∅ ∈ V)
4 0ss 4393 . . 3 ∅ ⊆ 𝑆
54a1i 11 . 2 (𝜑 → ∅ ⊆ 𝑆)
61, 3, 5nla0002 42010 1 (𝜑 → ∅ < ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  wss 3945  c0 4319   class class class wbr 5142  {copab 5204
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-xp 5676
This theorem is referenced by: (None)
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