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Mirrors > Home > MPE Home > Th. List > elreal | Structured version Visualization version GIF version |
Description: Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elreal | ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 10547 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 5579 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉) | |
4 | 0r 10502 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 3513 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 4804 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
7 | 6 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉)) |
8 | 5, 7 | rexsn 4620 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉) |
9 | eqcom 2828 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 0R〉 ↔ 〈𝑥, 0R〉 = 𝐴) | |
10 | 8, 9 | bitri 277 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 0R〉 = 𝐴) |
11 | 10 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
12 | 3, 11 | bitri 277 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
13 | 2, 12 | bitri 277 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {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: axaddrcl 10574 axmulrcl 10576 axrrecex 10585 axpre-lttri 10587 axpre-lttrn 10588 axpre-ltadd 10589 axpre-mulgt0 10590 |
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