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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 11046 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 5709 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 690 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 11098 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 8791 | . . 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 11091 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 11087 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2839 | 1 ⊢ 0R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 〈cop 4629 × cxp 5670 [cec 8721 / cqs 8722 Pcnp 10890 1Pc1p 10891 ~R cer 10895 Rcnr 10896 0Rc0r 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-inf2 9674 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-omul 8490 df-er 8723 df-ec 8725 df-qs 8729 df-ni 10903 df-pli 10904 df-mi 10905 df-lti 10906 df-plpq 10939 df-mpq 10940 df-ltpq 10941 df-enq 10942 df-nq 10943 df-erq 10944 df-plq 10945 df-mq 10946 df-1nq 10947 df-rq 10948 df-ltnq 10949 df-np 11012 df-1p 11013 df-enr 11086 df-nr 11087 df-0r 11091 |
This theorem is referenced by: sqgt0sr 11137 opelreal 11161 elreal 11162 elreal2 11163 eqresr 11168 addresr 11169 mulresr 11170 axresscn 11179 axicn 11181 axi2m1 11190 axcnre 11195 |
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