Theorem nla0001 44044

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 5272	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		0ss 4364	. . 3 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. 2 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6	1, 3, 5	nla0002 44042	1 ⊢ (𝜑 → ∅ < ∅)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113  {copab 5177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668

This theorem is referenced by: (None)