Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9677 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴) | |
2 | eleq1a 2683 | . . . . 5 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 +Q 𝑥) ∈ Q)) | |
3 | addnqf 9649 | . . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q | |
4 | 3 | fdmi 5965 | . . . . . . 7 ⊢ dom +Q = (Q × Q) |
5 | 0nnq 9625 | . . . . . . 7 ⊢ ¬ ∅ ∈ Q | |
6 | 4, 5 | ndmovrcl 6718 | . . . . . 6 ⊢ ((𝑥 +Q 𝑥) ∈ Q → (𝑥 ∈ Q ∧ 𝑥 ∈ Q)) |
7 | ltaddnq 9675 | . . . . . 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 34 | . . . 4 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q (𝑥 +Q 𝑥))) |
10 | breq2 4587 | . . . 4 ⊢ ((𝑥 +Q 𝑥) = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑥) ↔ 𝑥 <Q 𝐴)) | |
11 | 9, 10 | mpbidi 230 | . . 3 ⊢ (𝐴 ∈ Q → ((𝑥 +Q 𝑥) = 𝐴 → 𝑥 <Q 𝐴)) |
12 | 11 | eximdv 1833 | . 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 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 class class class wbr 4583 × cxp 5036 (class class class)co 6549 Qcnq 9553 +Q cplq 9556 <Q cltq 9559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 |
This theorem is referenced by: ltbtwnnq 9679 nqpr 9715 reclem2pr 9749 |
Copyright terms: Public domain | W3C validator |