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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 7263 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 4509 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 420 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 7433 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 6413 | . . 3 ⊢ (〈1P, 1P〉 ∈ (P × P) → [〈1P, 1P〉] ~R ∈ ((P × P) / ~R )) |
6 | 3, 5 | ax-mp 7 | . 2 ⊢ [〈1P, 1P〉] ~R ∈ ((P × P) / ~R ) |
7 | df-0r 7427 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 7423 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2181 | 1 ⊢ 0R ∈ R |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 〈cop 3477 × cxp 4475 [cec 6357 / cqs 6358 Pcnp 7000 1Pc1p 7001 ~R cer 7005 Rcnr 7006 0Rc0r 7007 |
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 df-enr 7422 df-nr 7423 df-0r 7427 |
This theorem is referenced by: addgt0sr 7471 ltadd1sr 7472 opelreal 7515 elreal 7516 elrealeu 7517 elreal2 7518 eqresr 7523 addresr 7524 mulresr 7525 pitonn 7535 peano2nnnn 7540 axresscn 7547 axicn 7550 axi2m1 7560 ax0id 7563 axprecex 7565 axcnre 7566 |
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