Theorem elreal 4044

Description: Membership in class of real numbers.

Assertion

Ref	Expression
elreal	⊢ (A ∈ ℝ ↔ ∃x(x ∈ R ∧ ⟨x, 0R⟩ = A))

Distinct variable group(s):   x,A

Proof of Theorem elreal

Step	Hyp	Ref	Expression
1		df-r 4038	. . 3 ⊢ ℝ = (R × {0R})
2	1	eleq2i 1153	. 2 ⊢ (A ∈ ℝ ↔ A ∈ (R × {0R}))
3		elxp 2442	. 2 ⊢ (A ∈ (R × {0R}) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})))
4		ancom 333	. . . . . . 7 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ ((x ∈ R ∧ y ∈ {0R}) ∧ A = ⟨x, y⟩))
5		anass 336	. . . . . . . 8 ⊢ (((x ∈ R ∧ y ∈ {0R}) ∧ A = ⟨x, y⟩) ↔ (x ∈ R ∧ (y ∈ {0R} ∧ A = ⟨x, y⟩)))
6		elsn 1820	. . . . . . . . . . 11 ⊢ (y ∈ {0R} ↔ y = 0R)
7		cleqcom 1103	. . . . . . . . . . 11 ⊢ (A = ⟨x, y⟩ ↔ ⟨x, y⟩ = A)
8	6, 7	anbi12i 369	. . . . . . . . . 10 ⊢ ((y ∈ {0R} ∧ A = ⟨x, y⟩) ↔ (y = 0R ∧ ⟨x, y⟩ = A))
9		opeq2 1877	. . . . . . . . . . . 12 ⊢ (y = 0R → ⟨x, y⟩ = ⟨x, 0R⟩)
10	9	cleq1d 1109	. . . . . . . . . . 11 ⊢ (y = 0R → (⟨x, y⟩ = A ↔ ⟨x, 0R⟩ = A))
11	10	pm5.32i 489	. . . . . . . . . 10 ⊢ ((y = 0R ∧ ⟨x, y⟩ = A) ↔ (y = 0R ∧ ⟨x, 0R⟩ = A))
12	8, 11	bitr 151	. . . . . . . . 9 ⊢ ((y ∈ {0R} ∧ A = ⟨x, y⟩) ↔ (y = 0R ∧ ⟨x, 0R⟩ = A))
13	12	anbi2i 367	. . . . . . . 8 ⊢ ((x ∈ R ∧ (y ∈ {0R} ∧ A = ⟨x, y⟩)) ↔ (x ∈ R ∧ (y = 0R ∧ ⟨x, 0R⟩ = A)))
14		an12 370	. . . . . . . 8 ⊢ ((x ∈ R ∧ (y = 0R ∧ ⟨x, 0R⟩ = A)) ↔ (y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
15	5, 13, 14	3bitr 155	. . . . . . 7 ⊢ (((x ∈ R ∧ y ∈ {0R}) ∧ A = ⟨x, y⟩) ↔ (y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
16	4, 15	bitr 151	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ (y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
17	16	biex 733	. . . . 5 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ ∃y(y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
18		19.41v 963	. . . . 5 ⊢ (∃y(y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)) ↔ (∃y y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
19	17, 18	bitr 151	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ (∃y y = 0R ∧ (x ∈ R ∧ ⟨x, 0R⟩ = A)))
20		0r 3983	. . . . . 6 ⊢ 0R ∈ R
21	20	elisseti 1355	. . . . 5 ⊢ 0R ∈ V
22	21	isseti 1352	. . . 4 ⊢ ∃y y = 0R
23	19, 22	mpbiran 547	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ (x ∈ R ∧ ⟨x, 0R⟩ = A))
24	23	biex 733	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ R ∧ y ∈ {0R})) ↔ ∃x(x ∈ R ∧ ⟨x, 0R⟩ = A))
25	2, 3, 24	3bitr 155	1 ⊢ (A ∈ ℝ ↔ ∃x(x ∈ R ∧ ⟨x, 0R⟩ = A))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  {csn 1808  ⟨cop 1810   × cxp 2408  Rcnr 3787  0_Rc0r 3788  ℝcr 4027

This theorem is referenced by:  suprelem 4053  supre 4054  ltsor 4055  axaddrcl 4067  axmulrcl 4069  axrnegex 4080  axrrecex 4081  axltadd 4085  axmulgt0 4086

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-enr 3960  df-nr 3961  df-0r 3965  df-r 4038

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