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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 11119 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2819 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 5693 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩) | |
4 | 0r 11074 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 3488 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 4869 | . . . . . . 7 ⊢ (𝑦 = 0R → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0R⟩) | |
7 | 6 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩)) |
8 | 5, 7 | rexsn 4681 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩) |
9 | eqcom 2733 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 0R⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) | |
10 | 8, 9 | bitri 275 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) |
11 | 10 | rexbii 3088 | . . 3 ⊢ (∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴) |
12 | 3, 11 | bitri 275 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴) |
13 | 2, 12 | bitri 275 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {csn 4623 ⟨cop 4629 × cxp 5667 Rcnr 10859 0Rc0r 10860 ℝcr 11108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-omul 8469 df-er 8702 df-ec 8704 df-qs 8708 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-mpq 10903 df-ltpq 10904 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-mq 10909 df-1nq 10910 df-rq 10911 df-ltnq 10912 df-np 10975 df-1p 10976 df-enr 11049 df-nr 11050 df-0r 11054 df-r 11119 |
This theorem is referenced by: axaddrcl 11146 axmulrcl 11148 axrrecex 11157 axpre-lttri 11159 axpre-lttrn 11160 axpre-ltadd 11161 axpre-mulgt0 11162 |
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