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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 10954 | . 2 ⊢ 1Q = 〈1o, 1o〉 | |
2 | 1pi 10921 | . . 3 ⊢ 1o ∈ N | |
3 | pinq 10965 | . . 3 ⊢ (1o ∈ N → 〈1o, 1o〉 ∈ Q) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈1o, 1o〉 ∈ Q |
5 | 1, 4 | eqeltri 2835 | 1 ⊢ 1Q ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 〈cop 4637 1oc1o 8498 Ncnpi 10882 Qcnq 10890 1Qc1q 10891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fv 6571 df-om 7888 df-2nd 8014 df-1o 8505 df-ni 10910 df-lti 10913 df-nq 10950 df-1nq 10954 |
This theorem is referenced by: nqerf 10968 mulidnq 11001 recmulnq 11002 recclnq 11004 1lt2nq 11011 halfnq 11014 1pr 11053 prlem934 11071 reclem3pr 11087 |
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