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Mirrors > Home > MPE Home > Th. List > 0r | Structured version Visualization version GIF version |
Description: The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0r | ⊢ 0R ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 10437 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 5592 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 690 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 10489 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 8353 | . . 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 10482 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 10478 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2926 | 1 ⊢ 0R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 〈cop 4573 × cxp 5553 [cec 8287 / cqs 8288 Pcnp 10281 1Pc1p 10282 ~R cer 10286 Rcnr 10287 0Rc0r 10288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-enr 10477 df-nr 10478 df-0r 10482 |
This theorem is referenced by: sqgt0sr 10528 opelreal 10552 elreal 10553 elreal2 10554 eqresr 10559 addresr 10560 mulresr 10561 axresscn 10570 axicn 10572 axi2m1 10581 axcnre 10586 |
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