![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ltrelre | GIF version |
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) |
Ref | Expression |
---|---|
ltrelre | ⊢ <ℝ ⊆ (ℝ × ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lt 7802 | . 2 ⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | |
2 | opabssxp 4696 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} ⊆ (ℝ × ℝ) | |
3 | 1, 2 | eqsstri 3187 | 1 ⊢ <ℝ ⊆ (ℝ × ℝ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ⊆ wss 3129 〈cop 3594 class class class wbr 4000 {copab 4060 × cxp 4620 0Rc0r 7275 <R cltr 7280 ℝcr 7788 <ℝ cltrr 7793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3135 df-ss 3142 df-opab 4062 df-xp 4628 df-lt 7802 |
This theorem is referenced by: ltresr 7816 |
Copyright terms: Public domain | W3C validator |