Theorem nla0002 43423

Description: Extending a linear order to subsets, the empty set is less than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)

Hypotheses

Ref	Expression
nla0001.defsslt	⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
nla0001.set	⊢ (𝜑 → 𝐴 ∈ V)
nla0002.sset	⊢ (𝜑 → 𝐴 ⊆ 𝑆)

Assertion

Ref	Expression
nla0002	⊢ (𝜑 → ∅ < 𝐴)

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 nla0002

Step	Hyp	Ref	Expression
1		0ex 5282	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
3		nla0001.set	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		0ss 4380	. . . 4 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
7		ral0 4493	. . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦
8	7	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)
9	5, 6, 8	3jca 1128	. 2 ⊢ (𝜑 → (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦))
10		nla0001.defsslt	. . 3 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
11	10	rp-brsslt 43422	. 2 ⊢ (∅ < 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)))
12	2, 3, 9, 11	syl21anbrc 1345	1 ⊢ (𝜑 → ∅ < 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ∅c0 4313   class class class wbr 5124  {copab 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665

This theorem is referenced by:  nla0001  43425