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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 7503 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 4641 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 424 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 7686 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 6563 | . . 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 7680 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 7676 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2252 | 1 ⊢ 0R ∈ R |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 〈cop 3584 × cxp 4607 [cec 6507 / cqs 6508 Pcnp 7240 1Pc1p 7241 ~R cer 7245 Rcnr 7246 0Rc0r 7247 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-pli 7254 df-mi 7255 df-lti 7256 df-plpq 7293 df-mpq 7294 df-enq 7296 df-nqqs 7297 df-plqqs 7298 df-mqqs 7299 df-1nqqs 7300 df-rq 7301 df-ltnqqs 7302 df-inp 7415 df-i1p 7416 df-enr 7675 df-nr 7676 df-0r 7680 |
This theorem is referenced by: addgt0sr 7724 ltadd1sr 7725 map2psrprg 7754 suplocsrlempr 7756 opelreal 7776 elreal 7777 elrealeu 7778 elreal2 7779 eqresr 7785 addresr 7786 mulresr 7787 pitonn 7797 peano2nnnn 7802 axresscn 7809 axicn 7812 axi2m1 7824 ax0id 7827 axprecex 7829 axcnre 7830 |
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