![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ltrelre | Structured version Visualization version GIF version |
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltrelre | ⊢ <ℝ ⊆ (ℝ × ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lt 11166 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
2 | opabssxp 5781 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ) | |
3 | 1, 2 | eqsstri 4030 | 1 ⊢ <ℝ ⊆ (ℝ × ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ⊆ wss 3963 〈cop 4637 class class class wbr 5148 {copab 5210 × cxp 5687 0Rc0r 10904 <R cltr 10909 ℝcr 11152 <ℝ cltrr 11157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-ss 3980 df-opab 5211 df-xp 5695 df-lt 11166 |
This theorem is referenced by: ltresr 11178 |
Copyright terms: Public domain | W3C validator |