Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10549 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 5581 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉) | |
4 | 0r 10504 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 3515 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 4806 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
7 | 6 | eqeq2d 2834 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉)) |
8 | 5, 7 | rexsn 4622 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉) |
9 | eqcom 2830 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 0R〉 ↔ 〈𝑥, 0R〉 = 𝐴) | |
10 | 8, 9 | bitri 277 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 0R〉 = 𝐴) |
11 | 10 | rexbii 3249 | . . 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 3141 {csn 4569 〈cop 4575 × cxp 5555 Rcnr 10289 0Rc0r 10290 ℝcr 10538 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-1p 10406 df-enr 10479 df-nr 10480 df-0r 10484 df-r 10549 |
This theorem is referenced by: axaddrcl 10576 axmulrcl 10578 axrrecex 10587 axpre-lttri 10589 axpre-lttrn 10590 axpre-ltadd 10591 axpre-mulgt0 10592 |
Copyright terms: Public domain | W3C validator |