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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 9738 | . 2 ⊢ 1Q = 〈1𝑜, 1𝑜〉 | |
2 | 1pi 9705 | . . 3 ⊢ 1𝑜 ∈ N | |
3 | pinq 9749 | . . 3 ⊢ (1𝑜 ∈ N → 〈1𝑜, 1𝑜〉 ∈ Q) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈1𝑜, 1𝑜〉 ∈ Q |
5 | 1, 4 | eqeltri 2697 | 1 ⊢ 1Q ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 〈cop 4183 1𝑜c1o 7553 Ncnpi 9666 Qcnq 9674 1Qc1q 9675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-om 7066 df-2nd 7169 df-1o 7560 df-ni 9694 df-lti 9697 df-nq 9734 df-1nq 9738 |
This theorem is referenced by: nqerf 9752 mulidnq 9785 recmulnq 9786 recclnq 9788 1lt2nq 9795 halfnq 9798 1pr 9837 prlem934 9855 reclem3pr 9871 |
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