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Mirrors > Home > ILE Home > Th. List > 1pr | GIF version |
Description: The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
Ref | Expression |
---|---|
1pr | ⊢ 1P ∈ P |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-i1p 7176 | . 2 ⊢ 1P = 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 | |
2 | 1nq 7075 | . . 3 ⊢ 1Q ∈ Q | |
3 | nqprlu 7256 | . . 3 ⊢ (1Q ∈ Q → 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 ∈ P) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 ∈ P |
5 | 1, 4 | eqeltri 2172 | 1 ⊢ 1P ∈ P |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 {cab 2086 〈cop 3477 class class class wbr 3875 Qcnq 6989 1Qc1q 6990 <Q cltq 6994 Pcnp 7000 1Pc1p 7001 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 df-inp 7175 df-i1p 7176 |
This theorem is referenced by: 1idprl 7299 1idpru 7300 1idpr 7301 recexprlemex 7346 ltmprr 7351 gt0srpr 7444 0r 7446 1sr 7447 m1r 7448 m1p1sr 7456 m1m1sr 7457 0lt1sr 7461 0idsr 7463 1idsr 7464 00sr 7465 recexgt0sr 7469 archsr 7477 srpospr 7478 prsrcl 7479 prsrpos 7480 prsradd 7481 prsrlt 7482 caucvgsrlembound 7489 pitonnlem1p1 7533 pitonnlem2 7534 pitonn 7535 pitoregt0 7536 pitore 7537 recnnre 7538 recidpirqlemcalc 7544 recidpirq 7545 |
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