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Mirrors > Home > ILE Home > Th. List > elprnql | GIF version |
Description: An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
elprnql | ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssnql 7135 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) | |
2 | 1 | sselda 3039 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1445 〈cop 3469 Qcnq 6936 Pcnp 6947 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-qs 6338 df-ni 6960 df-nqqs 7004 df-inp 7122 |
This theorem is referenced by: prubl 7142 prnmaxl 7144 prarloclemlt 7149 prarloclemlo 7150 prarloclem5 7156 genpdf 7164 genipv 7165 genpelvl 7168 genpml 7173 genprndl 7177 genpassl 7180 addnqprllem 7183 addnqprl 7185 addlocprlemeqgt 7188 addlocprlemgt 7190 addlocprlem 7191 nqprl 7207 prmuloc 7222 mulnqprl 7224 addcomprg 7234 mulcomprg 7236 distrlem1prl 7238 distrlem4prl 7240 1idprl 7246 ltsopr 7252 ltexprlemm 7256 ltexprlemopl 7257 ltexprlemopu 7259 ltexprlemupu 7260 ltexprlemdisj 7262 ltexprlemloc 7263 ltexprlemfl 7265 ltexprlemrl 7266 ltexprlemfu 7267 ltexprlemru 7268 addcanprleml 7270 addcanprlemu 7271 recexprlemloc 7287 recexprlem1ssl 7289 recexprlem1ssu 7290 recexprlemss1l 7291 aptiprleml 7295 aptiprlemu 7296 caucvgprprlemopl 7353 |
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