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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 7843 | . 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 4749 | . 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 3225 | 1 ⊢ <R ⊆ (R × R) |
| Colors of variables: wff set class |
| Syntax hints: ∧ wa 104 = wceq 1373 ∃wex 1515 ∈ wcel 2176 ⊆ wss 3166 〈cop 3636 class class class wbr 4044 {copab 4104 × cxp 4673 (class class class)co 5944 [cec 6618 +P cpp 7406 <P cltp 7408 ~R cer 7409 Rcnr 7410 <R cltr 7416 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-in 3172 df-ss 3179 df-opab 4106 df-xp 4681 df-ltr 7843 |
| This theorem is referenced by: gt0srpr 7861 recexgt0sr 7886 addgt0sr 7888 mulgt0sr 7891 caucvgsrlemcl 7902 caucvgsrlemasr 7903 caucvgsrlemfv 7904 map2psrprg 7918 suplocsrlemb 7919 suplocsrlempr 7920 suplocsrlem 7921 suplocsr 7922 ltresr 7952 axpre-ltirr 7995 axpre-lttrn 7997 |
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