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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 10771 | . . . 4 ⊢ 1P ∈ P | |
2 | opelxpi 5626 | . . . 4 ⊢ ((1P ∈ P ∧ 1P ∈ P) → 〈1P, 1P〉 ∈ (P × P)) | |
3 | 1, 1, 2 | mp2an 689 | . . 3 ⊢ 〈1P, 1P〉 ∈ (P × P) |
4 | enrex 10823 | . . . 4 ⊢ ~R ∈ V | |
5 | 4 | ecelqsi 8562 | . . 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 10816 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
8 | df-nr 10812 | . 2 ⊢ R = ((P × P) / ~R ) | |
9 | 6, 7, 8 | 3eltr4i 2852 | 1 ⊢ 0R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 〈cop 4567 × cxp 5587 [cec 8496 / cqs 8497 Pcnp 10615 1Pc1p 10616 ~R cer 10620 Rcnr 10621 0Rc0r 10622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-ec 8500 df-qs 8504 df-ni 10628 df-pli 10629 df-mi 10630 df-lti 10631 df-plpq 10664 df-mpq 10665 df-ltpq 10666 df-enq 10667 df-nq 10668 df-erq 10669 df-plq 10670 df-mq 10671 df-1nq 10672 df-rq 10673 df-ltnq 10674 df-np 10737 df-1p 10738 df-enr 10811 df-nr 10812 df-0r 10816 |
This theorem is referenced by: sqgt0sr 10862 opelreal 10886 elreal 10887 elreal2 10888 eqresr 10893 addresr 10894 mulresr 10895 axresscn 10904 axicn 10906 axi2m1 10915 axcnre 10920 |
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