Theorem nla0003 43387

Description: Extending a linear order to subsets, the empty set is greater 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
nla0003	⊢ (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 nla0003

Step	Hyp	Ref	Expression
1		nla0001.set	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		0ex 5325	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
5		0ss 4423	. . . 4 ⊢ ∅ ⊆ 𝑆
6	5	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
7		ral0 4536	. . . . 5 ⊢ ∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦
8		ralcom 3295	. . . . 5 ⊢ (∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦)
9	7, 8	mpbi 230	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦
10	9	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦)
11	4, 6, 10	3jca 1128	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦))
12		nla0001.defsslt	. . 3 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
13	12	rp-brsslt 43385	. 2 ⊢ (𝐴 < ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦)))
14	1, 3, 11, 13	syl21anbrc 1344	1 ⊢ (𝜑 → 𝐴 < ∅)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706

This theorem is referenced by: (None)