Theorem nla0003 42161

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 5306	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
5		0ss 4395	. . . 4 ⊢ ∅ ⊆ 𝑆
6	5	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
7		ral0 4511	. . . . 5 ⊢ ∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦
8		ralcom 3286	. . . . 5 ⊢ (∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦)
9	7, 8	mpbi 229	. . . 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 42159	. 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 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ⊆ wss 3947  ∅c0 4321   class class class wbr 5147  {copab 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681

This theorem is referenced by: (None)