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Theorem nla0003 41771
Description: Extending a linear order to subsets, the empty set is greater than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
Hypotheses
Ref Expression
nla0001.defsslt < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
nla0001.set (𝜑𝐴 ∈ V)
nla0002.sset (𝜑𝐴𝑆)
Assertion
Ref Expression
nla0003 (𝜑𝐴 < ∅)
Distinct variable groups:   𝐴,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 nla0003
StepHypRef Expression
1 nla0001.set . 2 (𝜑𝐴 ∈ V)
2 0ex 5269 . . 3 ∅ ∈ V
32a1i 11 . 2 (𝜑 → ∅ ∈ V)
4 nla0002.sset . . 3 (𝜑𝐴𝑆)
5 0ss 4361 . . . 4 ∅ ⊆ 𝑆
65a1i 11 . . 3 (𝜑 → ∅ ⊆ 𝑆)
7 ral0 4475 . . . . 5 𝑦 ∈ ∅ ∀𝑥𝐴 𝑥𝑅𝑦
8 ralcom 3275 . . . . 5 (∀𝑦 ∈ ∅ ∀𝑥𝐴 𝑥𝑅𝑦 ↔ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥𝑅𝑦)
97, 8mpbi 229 . . . 4 𝑥𝐴𝑦 ∈ ∅ 𝑥𝑅𝑦
109a1i 11 . . 3 (𝜑 → ∀𝑥𝐴𝑦 ∈ ∅ 𝑥𝑅𝑦)
114, 6, 103jca 1129 . 2 (𝜑 → (𝐴𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥𝑅𝑦))
12 nla0001.defsslt . . 3 < = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑆𝑏𝑆 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑅𝑦)}
1312rp-brsslt 41769 . 2 (𝐴 < ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥𝐴𝑦 ∈ ∅ 𝑥𝑅𝑦)))
141, 3, 11, 13syl21anbrc 1345 1 (𝜑𝐴 < ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  wss 3915  c0 4287   class class class wbr 5110  {copab 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644
This theorem is referenced by: (None)
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