Theorem nla0002 44686

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.defslts	⊢ < = {⟨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 5267	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
3		nla0001.set	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		0ss 4353	. . . 4 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
7		ral0 4457	. . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦
8	7	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)
9	5, 6, 8	3jca 1146	. 2 ⊢ (𝜑 → (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦))
10		nla0001.defslts	. . 3 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
11	10	rp-brsslt 44685	. 2 ⊢ (∅ < 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)))
12	2, 3, 9, 11	syl21anbrc 1363	1 ⊢ (𝜑 → ∅ < 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   class class class wbr 5108  {copab 5173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664

This theorem is used by:  nla0001  44688