Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . 2 ⊢ 0R = 0R | |
2 | df-r 10547 | . . . 4 ⊢ ℝ = (R × {0R}) | |
3 | 2 | eleq2i 2904 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 〈𝐴, 0R〉 ∈ (R × {0R})) |
4 | opelxp 5591 | . . 3 ⊢ (〈𝐴, 0R〉 ∈ (R × {0R}) ↔ (𝐴 ∈ R ∧ 0R ∈ {0R})) | |
5 | 0r 10502 | . . . . . 6 ⊢ 0R ∈ R | |
6 | 5 | elexi 3513 | . . . . 5 ⊢ 0R ∈ V |
7 | 6 | elsn 4582 | . . . 4 ⊢ (0R ∈ {0R} ↔ 0R = 0R) |
8 | 7 | anbi2i 624 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R ∈ {0R}) ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
9 | 3, 4, 8 | 3bitri 299 | . 2 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ (𝐴 ∈ R ∧ 0R = 0R)) |
10 | 1, 9 | mpbiran2 708 | 1 ⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4567 〈cop 4573 × cxp 5553 Rcnr 10287 0Rc0r 10288 ℝcr 10536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-enr 10477 df-nr 10478 df-0r 10482 df-r 10547 |
This theorem is referenced by: ltresr 10562 ax1cn 10571 axaddrcl 10574 axmulrcl 10576 axrnegex 10584 axrrecex 10585 axcnre 10586 axpre-sup 10591 |
Copyright terms: Public domain | W3C validator |