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 10707 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
2 | opabssxp 5625 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ) | |
3 | 1, 2 | eqsstri 3921 | 1 ⊢ <ℝ ⊆ (ℝ × ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ⊆ wss 3853 〈cop 4533 class class class wbr 5039 {copab 5101 × cxp 5534 0Rc0r 10445 <R cltr 10450 ℝcr 10693 <ℝ cltrr 10698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-in 3860 df-ss 3870 df-opab 5102 df-xp 5542 df-lt 10707 |
This theorem is referenced by: ltresr 10719 |
Copyright terms: Public domain | W3C validator |