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Mirrors > Home > MPE Home > Th. List > 1sr | Structured version Visualization version GIF version |
Description: The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1sr | ⊢ 1R ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 10431 | . . . . 5 ⊢ 1P ∈ P | |
2 | addclpr 10434 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 690 | . . . 4 ⊢ (1P +P 1P) ∈ P |
4 | opelxpi 5586 | . . . 4 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → 〈(1P +P 1P), 1P〉 ∈ (P × P)) | |
5 | 3, 1, 4 | mp2an 690 | . . 3 ⊢ 〈(1P +P 1P), 1P〉 ∈ (P × P) |
6 | enrex 10483 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 8347 | . . 3 ⊢ (〈(1P +P 1P), 1P〉 ∈ (P × P) → [〈(1P +P 1P), 1P〉] ~R ∈ ((P × P) / ~R )) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ [〈(1P +P 1P), 1P〉] ~R ∈ ((P × P) / ~R ) |
9 | df-1r 10477 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
10 | df-nr 10472 | . 2 ⊢ R = ((P × P) / ~R ) | |
11 | 8, 9, 10 | 3eltr4i 2926 | 1 ⊢ 1R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 〈cop 4566 × cxp 5547 (class class class)co 7150 [cec 8281 / cqs 8282 Pcnp 10275 1Pc1p 10276 +P cpp 10277 ~R cer 10280 Rcnr 10281 1Rc1r 10283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-1p 10398 df-plp 10399 df-enr 10471 df-nr 10472 df-1r 10477 |
This theorem is referenced by: 1ne0sr 10512 supsr 10528 ax1cn 10565 axicn 10566 axi2m1 10575 ax1ne0 10576 ax1rid 10577 axcnre 10580 |
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