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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nla0003 | Structured version Visualization version GIF version | ||
| 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.) | 
| 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 | 
|---|---|
| nla0003 | ⊢ (𝜑 → 𝐴 < ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nla0001.set | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
| 2 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ V) | 
| 4 | nla0002.sset | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
| 5 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ 𝑆 | |
| 6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ⊆ 𝑆) | 
| 7 | ral0 4513 | . . . . 5 ⊢ ∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦 | |
| 8 | ralcom 3289 | . . . . 5 ⊢ (∀𝑦 ∈ ∅ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦) | |
| 9 | 7, 8 | mpbi 230 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦 | 
| 10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦) | 
| 11 | 4, 6, 10 | 3jca 1129 | . 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 43436 | . 2 ⊢ (𝐴 < ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 ⊆ 𝑆 ∧ ∅ ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥𝑅𝑦))) | 
| 14 | 1, 3, 11, 13 | syl21anbrc 1345 | 1 ⊢ (𝜑 → 𝐴 < ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 {copab 5205 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 | 
| This theorem is referenced by: (None) | 
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