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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 10527 | . 2 ⊢ 1Q = 〈1o, 1o〉 | |
2 | 1pi 10494 | . . 3 ⊢ 1o ∈ N | |
3 | pinq 10538 | . . 3 ⊢ (1o ∈ N → 〈1o, 1o〉 ∈ Q) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈1o, 1o〉 ∈ Q |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 1Q ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 〈cop 4544 1oc1o 8192 Ncnpi 10455 Qcnq 10463 1Qc1q 10464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fv 6385 df-om 7642 df-2nd 7759 df-1o 8199 df-ni 10483 df-lti 10486 df-nq 10523 df-1nq 10527 |
This theorem is referenced by: nqerf 10541 mulidnq 10574 recmulnq 10575 recclnq 10577 1lt2nq 10584 halfnq 10587 1pr 10626 prlem934 10644 reclem3pr 10660 |
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