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Mirrors > Home > ILE Home > Th. List > 0r | GIF version |
Description: The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) |
Ref | Expression |
---|---|
0r | ⊢ 0R ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 7355 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 4566 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 422 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 7538 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 6476 | . . 3 ⊢ (〈1P, 1P〉 ∈ (P × P) → [〈1P, 1P〉] ~R ∈ ((P × P) / ~R )) |
6 | 3, 5 | ax-mp 5 | . 2 ⊢ [〈1P, 1P〉] ~R ∈ ((P × P) / ~R ) |
7 | df-0r 7532 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 7528 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2219 | 1 ⊢ 0R ∈ R |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 〈cop 3525 × cxp 4532 [cec 6420 / cqs 6421 Pcnp 7092 1Pc1p 7093 ~R cer 7097 Rcnr 7098 0Rc0r 7099 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-inp 7267 df-i1p 7268 df-enr 7527 df-nr 7528 df-0r 7532 |
This theorem is referenced by: addgt0sr 7576 ltadd1sr 7577 map2psrprg 7606 suplocsrlempr 7608 opelreal 7628 elreal 7629 elrealeu 7630 elreal2 7631 eqresr 7637 addresr 7638 mulresr 7639 pitonn 7649 peano2nnnn 7654 axresscn 7661 axicn 7664 axi2m1 7676 ax0id 7679 axprecex 7681 axcnre 7682 |
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