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Mirrors > Home > MPE Home > Th. List > Mathboxes > nla0002 | Structured version Visualization version GIF version |
Description: Extending a linear order to subsets, the empty set is less than any subset. 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)} |
nla0001.set | ⊢ (𝜑 → 𝐴 ∈ V) |
nla0002.sset | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
Ref | Expression |
---|---|
nla0002 | ⊢ (𝜑 → ∅ < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5269 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ V) |
3 | nla0001.set | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
4 | 0ss 4361 | . . . 4 ⊢ ∅ ⊆ 𝑆 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆) |
6 | nla0002.sset | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
7 | ral0 4475 | . . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦) |
9 | 5, 6, 8 | 3jca 1129 | . 2 ⊢ (𝜑 → (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦)) |
10 | nla0001.defsslt | . . 3 ⊢ < = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ 𝑆 ∧ 𝑏 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥𝑅𝑦)} | |
11 | 10 | rp-brsslt 41769 | . 2 ⊢ (∅ < 𝐴 ↔ ((∅ ∈ V ∧ 𝐴 ∈ V) ∧ (∅ ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆 ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ 𝐴 𝑥𝑅𝑦))) |
12 | 2, 3, 9, 11 | syl21anbrc 1345 | 1 ⊢ (𝜑 → ∅ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Vcvv 3448 ⊆ wss 3915 ∅c0 4287 class class class wbr 5110 {copab 5172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 |
This theorem is referenced by: nla0001 41772 |
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