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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 10769 | . . . . 5 ⊢ 1P ∈ P | |
2 | addclpr 10772 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 689 | . . . 4 ⊢ (1P +P 1P) ∈ P |
4 | opelxpi 5628 | . . . 4 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → 〈(1P +P 1P), 1P〉 ∈ (P × P)) | |
5 | 3, 1, 4 | mp2an 689 | . . 3 ⊢ 〈(1P +P 1P), 1P〉 ∈ (P × P) |
6 | enrex 10821 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 8560 | . . 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 10815 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
10 | df-nr 10810 | . 2 ⊢ R = ((P × P) / ~R ) | |
11 | 8, 9, 10 | 3eltr4i 2852 | 1 ⊢ 1R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 〈cop 4569 × cxp 5589 (class class class)co 7277 [cec 8494 / cqs 8495 Pcnp 10613 1Pc1p 10614 +P cpp 10615 ~R cer 10618 Rcnr 10619 1Rc1r 10621 |
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 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 |
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 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-oadd 8299 df-omul 8300 df-er 8496 df-ec 8498 df-qs 8502 df-ni 10626 df-pli 10627 df-mi 10628 df-lti 10629 df-plpq 10662 df-mpq 10663 df-ltpq 10664 df-enq 10665 df-nq 10666 df-erq 10667 df-plq 10668 df-mq 10669 df-1nq 10670 df-rq 10671 df-ltnq 10672 df-np 10735 df-1p 10736 df-plp 10737 df-enr 10809 df-nr 10810 df-1r 10815 |
This theorem is referenced by: 1ne0sr 10850 supsr 10866 ax1cn 10903 axicn 10904 axi2m1 10913 ax1ne0 10914 ax1rid 10915 axcnre 10918 |
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