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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 10475 | . 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 5638 | . 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 4001 | 1 ⊢ <R ⊆ (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ⊆ wss 3936 〈cop 4567 class class class wbr 5059 {copab 5121 × cxp 5548 (class class class)co 7150 [cec 8281 +P cpp 10277 <P cltp 10279 ~R cer 10280 Rcnr 10281 <R cltr 10287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-opab 5122 df-xp 5556 df-ltr 10475 |
This theorem is referenced by: ltsrpr 10493 ltasr 10516 recexsrlem 10519 addgt0sr 10520 mulgt0sr 10521 map2psrpr 10526 supsrlem 10527 supsr 10528 ltresr 10556 axpre-lttrn 10582 |
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