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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 11068 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 5662 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩) | |
4 | 0r 11023 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 3467 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 4836 | . . . . . . 7 ⊢ (𝑦 = 0R → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0R⟩) | |
7 | 6 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩)) |
8 | 5, 7 | rexsn 4648 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩) |
9 | eqcom 2744 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 0R⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) | |
10 | 8, 9 | bitri 275 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) |
11 | 10 | rexbii 3098 | . . 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 1542 ∈ wcel 2107 ∃wrex 3074 {csn 4591 ⟨cop 4597 × cxp 5636 Rcnr 10808 0Rc0r 10809 ℝcr 11057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-1p 10925 df-enr 10998 df-nr 10999 df-0r 11003 df-r 11068 |
This theorem is referenced by: axaddrcl 11095 axmulrcl 11097 axrrecex 11106 axpre-lttri 11108 axpre-lttrn 11109 axpre-ltadd 11110 axpre-mulgt0 11111 |
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