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Mirrors > Home > ILE Home > Th. List > nsmallnq | GIF version |
Description: There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
nsmallnq | ⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsmallnqq 7430 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴) | |
2 | rexex 2536 | . 2 ⊢ (∃𝑥 ∈ Q 𝑥 <Q 𝐴 → ∃𝑥 𝑥 <Q 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1503 ∈ wcel 2160 ∃wrex 2469 class class class wbr 4018 Qcnq 7298 <Q cltq 7303 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-1o 6435 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7322 df-pli 7323 df-mi 7324 df-lti 7325 df-plpq 7362 df-mpq 7363 df-enq 7365 df-nqqs 7366 df-plqqs 7367 df-mqqs 7368 df-1nqqs 7369 df-rq 7370 df-ltnqqs 7371 |
This theorem is referenced by: ltbtwnnqq 7433 |
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