Theorem nla0003 42665

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 5297	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		nla0002.sset	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
5		0ss 4388	. . . 4 ⊢ ∅ ⊆ 𝑆
6	5	a1i 11	. . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆)
7		ral0 4504	. . . . 5 ⊢ ∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦
8		ralcom 3278	. . . . 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 1125	. 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 42663	. 2 ⊢ (𝐴 < ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦)))
14	1, 3, 11, 13	syl21anbrc 1341	1 ⊢ (𝜑 → 𝐴 < ∅)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ⊆ wss 3940  ∅c0 4314   class class class wbr 5138  {copab 5200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672

This theorem is referenced by: (None)