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Mirrors > Home > MPE Home > Th. List > opelreal | Structured version Visualization version GIF version |
Description: Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opelreal | ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . 2 ⊢ 0R = 0R | |
2 | df-r 10870 | . . . 4 ⊢ ℝ = (R × {0R}) | |
3 | 2 | eleq2i 2830 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 〈𝐴, 0R〉 ∈ (R × {0R})) |
4 | opelxp 5622 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ (R × {0R}) ↔ (𝐴 ∈ R ∧ 0R ∈ {0R})) | |
5 | 0r 10825 | . . . . . 6 ⊢ 0R ∈ R | |
6 | 5 | elexi 3450 | . . . . 5 ⊢ 0R ∈ V |
7 | 6 | elsn 4578 | . . . 4 ⊢ (0R ∈ {0R} ↔ 0R = 0R) |
8 | 7 | anbi2i 623 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R ∈ {0R}) ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
9 | 3, 4, 8 | 3bitri 297 | . 2 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
10 | 1, 9 | mpbiran2 707 | 1 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {csn 4563 〈cop 4569 × cxp 5584 Rcnr 10610 0Rc0r 10611 ℝcr 10859 |
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 7580 ax-inf2 9388 |
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 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-omul 8291 df-er 8487 df-ec 8489 df-qs 8493 df-ni 10617 df-pli 10618 df-mi 10619 df-lti 10620 df-plpq 10653 df-mpq 10654 df-ltpq 10655 df-enq 10656 df-nq 10657 df-erq 10658 df-plq 10659 df-mq 10660 df-1nq 10661 df-rq 10662 df-ltnq 10663 df-np 10726 df-1p 10727 df-enr 10800 df-nr 10801 df-0r 10805 df-r 10870 |
This theorem is referenced by: ltresr 10885 ax1cn 10894 axaddrcl 10897 axmulrcl 10899 axrnegex 10907 axrrecex 10908 axcnre 10909 axpre-sup 10914 |
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