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Mirrors > Home > ILE Home > Th. List > ltrelsr | GIF version |
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Ref | Expression |
---|---|
ltrelsr | ⊢ <R ⊆ (R × R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltr 7538 | . 2 ⊢ <R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ~R ∧ 𝑦 = [〈𝑣, 𝑢〉] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} | |
2 | opabssxp 4613 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ~R ∧ 𝑦 = [〈𝑣, 𝑢〉] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} ⊆ (R × R) | |
3 | 1, 2 | eqsstri 3129 | 1 ⊢ <R ⊆ (R × R) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3071 〈cop 3530 class class class wbr 3929 {copab 3988 × cxp 4537 (class class class)co 5774 [cec 6427 +P cpp 7101 <P cltp 7103 ~R cer 7104 Rcnr 7105 <R cltr 7111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-in 3077 df-ss 3084 df-opab 3990 df-xp 4545 df-ltr 7538 |
This theorem is referenced by: gt0srpr 7556 recexgt0sr 7581 addgt0sr 7583 mulgt0sr 7586 caucvgsrlemcl 7597 caucvgsrlemasr 7598 caucvgsrlemfv 7599 map2psrprg 7613 suplocsrlemb 7614 suplocsrlempr 7615 suplocsrlem 7616 suplocsr 7617 ltresr 7647 axpre-ltirr 7690 axpre-lttrn 7692 |
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