Theorem nla0002 44007

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 5261	. . 3 ⊢ ∅ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
3		nla0001.set	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		0ss 4357	. . . 4 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
7		ral0 4455	. . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦
8	7	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)
9	5, 6, 8	3jca 1144	. 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 44006	. 2 ⊢ (∅ < 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)))
12	2, 3, 9, 11	syl21anbrc 1361	1 ⊢ (𝜑 → ∅ < 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  ∅c0 4288   class class class wbr 5104  {copab 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657

This theorem is referenced by:  nla0001  44009