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Mirrors > Home > MPE Home > Th. List > nsmallnq | Structured version Visualization version GIF version |
Description: The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nsmallnq | ⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnq 10386 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴) | |
2 | eleq1a 2905 | . . . . 5 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 +Q 𝑥) ∈ Q)) | |
3 | addnqf 10358 | . . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q | |
4 | 3 | fdmi 6517 | . . . . . . 7 ⊢ dom +Q = (Q × Q) |
5 | 0nnq 10334 | . . . . . . 7 ⊢ ¬ ∅ ∈ Q | |
6 | 4, 5 | ndmovrcl 7323 | . . . . . 6 ⊢ ((𝑥 +Q 𝑥) ∈ Q → (𝑥 ∈ Q ∧ 𝑥 ∈ Q)) |
7 | ltaddnq 10384 | . . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑥)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝑥 +Q 𝑥) ∈ Q → 𝑥 <Q (𝑥 +Q 𝑥)) |
9 | 2, 8 | syl6 35 | . . . 4 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q (𝑥 +Q 𝑥))) |
10 | breq2 5061 | . . . 4 ⊢ ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑥) ↔ 𝑥 <Q 𝐴)) | |
11 | 9, 10 | mpbidi 242 | . . 3 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q 𝐴)) |
12 | 11 | eximdv 1909 | . 2 ⊢ (𝐴 ∈ Q → (∃𝑥(𝑥 +Q 𝑥) = 𝐴 → ∃𝑥 𝑥 <Q 𝐴)) |
13 | 1, 12 | mpd 15 | 1 ⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 class class class wbr 5057 × cxp 5546 (class class class)co 7145 Qcnq 10262 +Q cplq 10265 <Q cltq 10268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-rq 10327 df-ltnq 10328 |
This theorem is referenced by: ltbtwnnq 10388 nqpr 10424 reclem2pr 10458 |
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