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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 10194 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴) | |
2 | eleq1a 2855 | . . . . 5 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 +Q 𝑥) ∈ Q)) | |
3 | addnqf 10166 | . . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q | |
4 | 3 | fdmi 6351 | . . . . . . 7 ⊢ dom +Q = (Q × Q) |
5 | 0nnq 10142 | . . . . . . 7 ⊢ ¬ ∅ ∈ Q | |
6 | 4, 5 | ndmovrcl 7148 | . . . . . 6 ⊢ ((𝑥 +Q 𝑥) ∈ Q → (𝑥 ∈ Q ∧ 𝑥 ∈ Q)) |
7 | ltaddnq 10192 | . . . . . 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 4929 | . . . 4 ⊢ ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑥) ↔ 𝑥 <Q 𝐴)) | |
11 | 9, 10 | mpbidi 233 | . . 3 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q 𝐴)) |
12 | 11 | eximdv 1876 | . 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 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 class class class wbr 4925 × cxp 5401 (class class class)co 6974 Qcnq 10070 +Q cplq 10073 <Q cltq 10076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-omul 7908 df-er 8087 df-ni 10090 df-pli 10091 df-mi 10092 df-lti 10093 df-plpq 10126 df-mpq 10127 df-ltpq 10128 df-enq 10129 df-nq 10130 df-erq 10131 df-plq 10132 df-mq 10133 df-1nq 10134 df-rq 10135 df-ltnq 10136 |
This theorem is referenced by: ltbtwnnq 10196 nqpr 10232 reclem2pr 10266 |
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