Theorem nla0001 43870

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.defslts	⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 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

Step	Hyp	Ref	Expression
1		nla0001.defslts	. 2 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
2		0ex 5229	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		0ss 4328	. . 3 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. 2 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6	1, 3, 5	nla0002 43868	1 ⊢ (𝜑 → ∅ < ∅)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  ∅c0 4261   class class class wbr 5072  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624

This theorem is referenced by: (None)