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| Mirrors > Home > MPE Home > Th. List > ltrelsr | Structured version Visualization version GIF version | ||
| Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ltrelsr | ⊢ <R ⊆ (R × R) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ltr 11099 | . 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 5778 | . 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 4030 | 1 ⊢ <R ⊆ (R × R) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 (class class class)co 7431 [cec 8743 +P cpp 10901 <P cltp 10903 ~R cer 10904 Rcnr 10905 <R cltr 10911 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ss 3968 df-opab 5206 df-xp 5691 df-ltr 11099 |
| This theorem is referenced by: ltsrpr 11117 ltasr 11140 recexsrlem 11143 addgt0sr 11144 mulgt0sr 11145 map2psrpr 11150 supsrlem 11151 supsr 11152 ltresr 11180 axpre-lttrn 11206 |
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