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Mirrors > Home > MPE Home > Th. List > 1nq | Structured version Visualization version GIF version |
Description: The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1nq | ⊢ 1Q ∈ Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1nq 10326 | . 2 ⊢ 1Q = 〈1o, 1o〉 | |
2 | 1pi 10293 | . . 3 ⊢ 1o ∈ N | |
3 | pinq 10337 | . . 3 ⊢ (1o ∈ N → 〈1o, 1o〉 ∈ Q) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈1o, 1o〉 ∈ Q |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 1Q ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 〈cop 4563 1oc1o 8084 Ncnpi 10254 Qcnq 10262 1Qc1q 10263 |
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-rab 3144 df-v 3494 df-sbc 3770 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-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-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-om 7570 df-2nd 7679 df-1o 8091 df-ni 10282 df-lti 10285 df-nq 10322 df-1nq 10326 |
This theorem is referenced by: nqerf 10340 mulidnq 10373 recmulnq 10374 recclnq 10376 1lt2nq 10383 halfnq 10386 1pr 10425 prlem934 10443 reclem3pr 10459 |
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