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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 2779 | . 2 ⊢ 0R = 0R | |
2 | df-r 10345 | . . . 4 ⊢ ℝ = (R × {0R}) | |
3 | 2 | eleq2i 2858 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 〈𝐴, 0R〉 ∈ (R × {0R})) |
4 | opelxp 5443 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ (R × {0R}) ↔ (𝐴 ∈ R ∧ 0R ∈ {0R})) | |
5 | 0r 10300 | . . . . . 6 ⊢ 0R ∈ R | |
6 | 5 | elexi 3435 | . . . . 5 ⊢ 0R ∈ V |
7 | 6 | elsn 4456 | . . . 4 ⊢ (0R ∈ {0R} ↔ 0R = 0R) |
8 | 7 | anbi2i 613 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R ∈ {0R}) ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
9 | 3, 4, 8 | 3bitri 289 | . 2 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
10 | 1, 9 | mpbiran2 697 | 1 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {csn 4441 〈cop 4447 × cxp 5405 Rcnr 10085 0Rc0r 10086 ℝcr 10334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-omul 7910 df-er 8089 df-ec 8091 df-qs 8095 df-ni 10092 df-pli 10093 df-mi 10094 df-lti 10095 df-plpq 10128 df-mpq 10129 df-ltpq 10130 df-enq 10131 df-nq 10132 df-erq 10133 df-plq 10134 df-mq 10135 df-1nq 10136 df-rq 10137 df-ltnq 10138 df-np 10201 df-1p 10202 df-enr 10275 df-nr 10276 df-0r 10280 df-r 10345 |
This theorem is referenced by: ltresr 10360 ax1cn 10369 axaddrcl 10372 axmulrcl 10374 axrnegex 10382 axrrecex 10383 axcnre 10384 axpre-sup 10389 |
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