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| 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 11168 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
| 2 | opabssxp 5778 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 0Rc0r 10906 <R cltr 10911 ℝcr 11154 <ℝ cltrr 11159 | 
| 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-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ss 3968 df-opab 5206 df-xp 5691 df-lt 11168 | 
| This theorem is referenced by: ltresr 11180 | 
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