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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 7299 | . 2 ⊢ 1P = 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 | |
2 | 1nq 7198 | . . 3 ⊢ 1Q ∈ Q | |
3 | nqprlu 7379 | . . 3 ⊢ (1Q ∈ Q → 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 ∈ P) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 ∈ P |
5 | 1, 4 | eqeltri 2213 | 1 ⊢ 1P ∈ P |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 {cab 2126 〈cop 3535 class class class wbr 3937 Qcnq 7112 1Qc1q 7113 <Q cltq 7117 Pcnp 7123 1Pc1p 7124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-inp 7298 df-i1p 7299 |
This theorem is referenced by: 1idprl 7422 1idpru 7423 1idpr 7424 recexprlemex 7469 ltmprr 7474 gt0srpr 7580 0r 7582 1sr 7583 m1r 7584 m1p1sr 7592 m1m1sr 7593 0lt1sr 7597 0idsr 7599 1idsr 7600 00sr 7601 recexgt0sr 7605 archsr 7614 srpospr 7615 prsrcl 7616 prsrpos 7617 prsradd 7618 prsrlt 7619 caucvgsrlembound 7626 ltpsrprg 7635 mappsrprg 7636 map2psrprg 7637 suplocsrlemb 7638 suplocsrlempr 7639 pitonnlem1p1 7678 pitonnlem2 7679 pitonn 7680 pitoregt0 7681 pitore 7682 recnnre 7683 recidpirqlemcalc 7689 recidpirq 7690 |
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