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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 10400 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴) | |
2 | eleq1a 2910 | . . . . 5 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 +Q 𝑥) ∈ Q)) | |
3 | addnqf 10372 | . . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q | |
4 | 3 | fdmi 6526 | . . . . . . 7 ⊢ dom +Q = (Q × Q) |
5 | 0nnq 10348 | . . . . . . 7 ⊢ ¬ ∅ ∈ Q | |
6 | 4, 5 | ndmovrcl 7336 | . . . . . 6 ⊢ ((𝑥 +Q 𝑥) ∈ Q → (𝑥 ∈ Q ∧ 𝑥 ∈ Q)) |
7 | ltaddnq 10398 | . . . . . 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 5072 | . . . 4 ⊢ ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑥) ↔ 𝑥 <Q 𝐴)) | |
11 | 9, 10 | mpbidi 243 | . . 3 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q 𝐴)) |
12 | 11 | eximdv 1918 | . 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 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 class class class wbr 5068 × cxp 5555 (class class class)co 7158 Qcnq 10276 +Q cplq 10279 <Q cltq 10282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 |
This theorem is referenced by: ltbtwnnq 10402 nqpr 10438 reclem2pr 10472 |
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