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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 10264 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
2 | opabssxp 5427 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ) | |
3 | 1, 2 | eqsstri 3859 | 1 ⊢ <ℝ ⊆ (ℝ × ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ⊆ wss 3797 〈cop 4402 class class class wbr 4872 {copab 4934 × cxp 5339 0Rc0r 10002 <R cltr 10007 ℝcr 10250 <ℝ cltrr 10255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-in 3804 df-ss 3811 df-opab 4935 df-xp 5347 df-lt 10264 |
This theorem is referenced by: ltresr 10276 |
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