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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 11149 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2821 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 5702 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩) | |
4 | 0r 11104 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 3491 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 4875 | . . . . . . 7 ⊢ (𝑦 = 0R → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0R⟩) | |
7 | 6 | eqeq2d 2739 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩)) |
8 | 5, 7 | rexsn 4687 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 0R⟩) |
9 | eqcom 2735 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 0R⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) | |
10 | 8, 9 | bitri 275 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 0R⟩ = 𝐴) |
11 | 10 | rexbii 3091 | . . 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 1534 ∈ wcel 2099 ∃wrex 3067 {csn 4629 ⟨cop 4635 × cxp 5676 Rcnr 10889 0Rc0r 10890 ℝcr 11138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10896 df-pli 10897 df-mi 10898 df-lti 10899 df-plpq 10932 df-mpq 10933 df-ltpq 10934 df-enq 10935 df-nq 10936 df-erq 10937 df-plq 10938 df-mq 10939 df-1nq 10940 df-rq 10941 df-ltnq 10942 df-np 11005 df-1p 11006 df-enr 11079 df-nr 11080 df-0r 11084 df-r 11149 |
This theorem is referenced by: axaddrcl 11176 axmulrcl 11178 axrrecex 11187 axpre-lttri 11189 axpre-lttrn 11190 axpre-ltadd 11191 axpre-mulgt0 11192 |
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