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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 9617 | . 2 ⊢ 1Q = 〈1𝑜, 1𝑜〉 | |
2 | 1pi 9584 | . . 3 ⊢ 1𝑜 ∈ N | |
3 | pinq 9628 | . . 3 ⊢ (1𝑜 ∈ N → 〈1𝑜, 1𝑜〉 ∈ Q) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈1𝑜, 1𝑜〉 ∈ Q |
5 | 1, 4 | eqeltri 2684 | 1 ⊢ 1Q ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 〈cop 4131 1𝑜c1o 7440 Ncnpi 9545 Qcnq 9553 1Qc1q 9554 |
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-rab 2905 df-v 3175 df-sbc 3403 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-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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fv 5812 df-om 6958 df-2nd 7060 df-1o 7447 df-ni 9573 df-lti 9576 df-nq 9613 df-1nq 9617 |
This theorem is referenced by: nqerf 9631 mulidnq 9664 recmulnq 9665 recclnq 9667 1lt2nq 9674 halfnq 9677 1pr 9716 prlem934 9734 reclem3pr 9750 |
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