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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 10544 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
2 | opabssxp 5637 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ) | |
3 | 1, 2 | eqsstri 4000 | 1 ⊢ <ℝ ⊆ (ℝ × ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ⊆ wss 3935 〈cop 4566 class class class wbr 5058 {copab 5120 × cxp 5547 0Rc0r 10282 <R cltr 10287 ℝcr 10530 <ℝ cltrr 10535 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3942 df-ss 3951 df-opab 5121 df-xp 5555 df-lt 10544 |
This theorem is referenced by: ltresr 10556 |
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