Theorem nla0002

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)

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

Proof

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