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Mirrors > Home > MPE Home > Th. List > Mathboxes > nla0001 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nla0001.defsslt | ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} |
Ref | Expression |
---|---|
nla0001 | ⊢ (𝜑 → ∅ < ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nla0001.defsslt | . 2 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} | |
2 | 0ex 5309 | . . 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 42857 | 1 ⊢ (𝜑 → ∅ < ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3057 Vcvv 3471 ⊆ wss 3947 ∅c0 4324 class class class wbr 5150 {copab 5212 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
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 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5151 df-opab 5213 df-xp 5686 |
This theorem is referenced by: (None) |
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