Theorem nla0001 42965

Description: Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)

Hypothesis

Ref	Expression
nla0001.defsslt	⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}

Assertion

Ref	Expression
nla0001	⊢ (𝜑 → ∅ < ∅)

Distinct variable groups:   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 nla0001

Step	Hyp	Ref	Expression
1		nla0001.defsslt	. 2 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)}
2		0ex 5308	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → ∅ ∈ V)
4		0ss 4398	. . 3 ⊢ ∅ ⊆ 𝑆
5	4	a1i 11	. 2 ⊢ (𝜑 → ∅ ⊆ 𝑆)
6	1, 3, 5	nla0002 42963	1 ⊢ (𝜑 → ∅ < ∅)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050  Vcvv 3461   ⊆ wss 3944  ∅c0 4322   class class class wbr 5149  {copab 5211

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-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684

This theorem is referenced by: (None)

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