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Theorem elreal2 11092

Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
elreal2	⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))

**Proof of Theorem elreal2**
Step	Hyp	Ref	Expression
1		df-r 11085	. . 3 ⊢ ℝ = (R × {0_R})
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0_R}))
3		xp1st 8003	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → (1^st ‘𝐴) ∈ R)
4		1st2nd2 8010	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5		xp2nd 8004	. . . . . . 7 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) ∈ {0_R})
6		elsni 4609	. . . . . . 7 ⊢ ((2^nd ‘𝐴) ∈ {0_R} → (2^nd ‘𝐴) = 0_R)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (R × {0_R}) → (2^nd ‘𝐴) = 0_R)
8	7	opeq2d 4847	. . . . 5 ⊢ (𝐴 ∈ (R × {0_R}) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐴), 0_R⟩)
9	4, 8	eqtrd 2765	. . . 4 ⊢ (𝐴 ∈ (R × {0_R}) → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
10	3, 9	jca 511	. . 3 ⊢ (𝐴 ∈ (R × {0_R}) → ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
11		eleq1 2817	. . . . 5 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ ⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R})))
12		0r 11040	. . . . . . . 8 ⊢ 0_R ∈ R
13	12	elexi 3473	. . . . . . 7 ⊢ 0_R ∈ V
14	13	snid 4629	. . . . . 6 ⊢ 0_R ∈ {0_R}
15		opelxp 5677	. . . . . 6 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 0_R ∈ {0_R}))
16	14, 15	mpbiran2 710	. . . . 5 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R)
17	11, 16	bitrdi 287	. . . 4 ⊢ (𝐴 = ⟨(1^st ‘𝐴), 0_R⟩ → (𝐴 ∈ (R × {0_R}) ↔ (1^st ‘𝐴) ∈ R))
18	17	biimparc 479	. . 3 ⊢ (((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩) → 𝐴 ∈ (R × {0_R}))
19	10, 18	impbii 209	. 2 ⊢ (𝐴 ∈ (R × {0_R}) ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
20	2, 19	bitri 275	1 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {csn 4592 ⟨cop 4598 × cxp 5639 ‘cfv 6514 1^st c1st 7969 2^nd c2nd 7970 Rcnr 10825 0_Rc0r 10826 ℝcr 11074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-omul 8442 df-er 8674 df-ec 8676 df-qs 8680 df-ni 10832 df-pli 10833 df-mi 10834 df-lti 10835 df-plpq 10868 df-mpq 10869 df-ltpq 10870 df-enq 10871 df-nq 10872 df-erq 10873 df-plq 10874 df-mq 10875 df-1nq 10876 df-rq 10877 df-ltnq 10878 df-np 10941 df-1p 10942 df-enr 11015 df-nr 11016 df-0r 11020 df-r 11085

This theorem is referenced by: ltresr2 11101 axrnegex 11122 axpre-sup 11129

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