arch - Metamath Proof Explorer

Metamath Proof Explorer

< Previous Next >
Related theorems
GIF version

Theorem arch 6018

Description: Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26.

**Assertion**
Ref	Expression
arch	⊢ (A ∈ ℝ → ∃n ∈ ℕ A < n)

Distinct variable group: A,n

**Proof of Theorem arch**
Step	Hyp	Ref	Expression
1		breq1 2612	. . 3 ⊢ (y = A → (y < n ↔ A < n))
2	1	rexbidv 1656	. 2 ⊢ (y = A → (∃n ∈ ℕ y < n ↔ ∃n ∈ ℕ A < n))
3		nnunb 6017	. . . 4 ⊢ ¬ ∃y ∈ ℝ ∀n ∈ ℕ (n < y ⋁ n = y)
4		ralnex 1645	. . . 4 ⊢ (∀y ∈ ℝ ¬ ∀n ∈ ℕ (n < y ⋁ n = y) ↔ ¬ ∃y ∈ ℝ ∀n ∈ ℕ (n < y ⋁ n = y))
5	3, 4	mpbir 190	. . 3 ⊢ ∀y ∈ ℝ ¬ ∀n ∈ ℕ (n < y ⋁ n = y)
6		axlttri 5475	. . . . . . . . 9 ⊢ ((y ∈ ℝ ⋀ n ∈ ℝ) → (y < n ↔ ¬ (y = n ⋁ n < y)))
7		nnret 5877	. . . . . . . . 9 ⊢ (n ∈ ℕ → n ∈ ℝ)
8	6, 7	sylan2 451	. . . . . . . 8 ⊢ ((y ∈ ℝ ⋀ n ∈ ℕ) → (y < n ↔ ¬ (y = n ⋁ n < y)))
9		eqcom 1469	. . . . . . . . . . 11 ⊢ (y = n ↔ n = y)
10	9	orbi1i 256	. . . . . . . . . 10 ⊢ ((y = n ⋁ n < y) ↔ (n = y ⋁ n < y))
11		orcom 246	. . . . . . . . . 10 ⊢ ((n = y ⋁ n < y) ↔ (n < y ⋁ n = y))
12	10, 11	bitr 173	. . . . . . . . 9 ⊢ ((y = n ⋁ n < y) ↔ (n < y ⋁ n = y))
13	12	negbii 187	. . . . . . . 8 ⊢ (¬ (y = n ⋁ n < y) ↔ ¬ (n < y ⋁ n = y))
14	8, 13	syl6bb 534	. . . . . . 7 ⊢ ((y ∈ ℝ ⋀ n ∈ ℕ) → (y < n ↔ ¬ (n < y ⋁ n = y)))
15	14	biimprd 154	. . . . . 6 ⊢ ((y ∈ ℝ ⋀ n ∈ ℕ) → (¬ (n < y ⋁ n = y) → y < n))
16	15	r19.22dva 1731	. . . . 5 ⊢ (y ∈ ℝ → (∃n ∈ ℕ ¬ (n < y ⋁ n = y) → ∃n ∈ ℕ y < n))
17		rexnal 1646	. . . . 5 ⊢ (∃n ∈ ℕ ¬ (n < y ⋁ n = y) ↔ ¬ ∀n ∈ ℕ (n < y ⋁ n = y))
18	16, 17	syl5ibr 207	. . . 4 ⊢ (y ∈ ℝ → (¬ ∀n ∈ ℕ (n < y ⋁ n = y) → ∃n ∈ ℕ y < n))
19	18	r19.20i 1696	. . 3 ⊢ (∀y ∈ ℝ ¬ ∀n ∈ ℕ (n < y ⋁ n = y) → ∀y ∈ ℝ ∃n ∈ ℕ y < n)
20	5, 19	ax-mp 7	. 2 ⊢ ∀y ∈ ℝ ∃n ∈ ℕ y < n
21	2, 20	vtoclri 1850	1 ⊢ (A ∈ ℝ → ∃n ∈ ℕ A < n)

Colors of variables: wff set class

Syntax hints: ¬ wn 2 → wi 3 ↔ wb 146 ⋁ wo 222 ⋀ wa 223 = wceq 953 ∈ wcel 955 ∀wral 1637 ∃wrex 1638 class class class wbr 2609 ℝcr 5205 ℕcn 5268 < clt 5458

This theorem is referenced by: nnreclt 6019 bndndx 6020 btwnz 6163 ubthlem5 8464 projlem1 9102 projlem26 9127

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-9 962 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-rep 2683 ax-sep 2693 ax-nul 2700 ax-pow 2732 ax-pr 2769 ax-un 2857 ax-inf2 4597

This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-nel 1580 df-ral 1641 df-rex 1642 df-reu 1643 df-rab 1644 df-v 1803 df-sbc 1932 df-csb 1992 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-pss 2045 df-nul 2271 df-if 2352 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-int 2524 df-iun 2558 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-id 2824 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-suc 2944 df-om 3122 df-xp 3174 df-rel 3175 df-cnv 3176 df-co 3177 df-dm 3178 df-rn 3179 df-res 3180 df-ima 3181 df-fun 3182 df-fn 3183 df-f 3184 df-f1 3185 df-fo 3186 df-f1o 3187 df-fv 3188 df-rdg 3917 df-opr 3950 df-oprab 3951 df-1st 4063 df-2nd 4064 df-1o 4117 df-oadd 4119 df-omul 4120 df-er 4245 df-ec 4247 df-qs 4250 df-en 4351 df-dom 4352 df-sdom 4353 df-ni 4972 df-pli 4973 df-mi 4974 df-lti 4975 df-plpq 5007 df-mpq 5008 df-enq 5009 df-nq 5010 df-plq 5011 df-mq 5012 df-rq 5013 df-ltq 5014 df-1q 5015 df-np 5058 df-1p 5059 df-plp 5060 df-mp 5061 df-ltp 5062 df-plpr 5136 df-mpr 5137 df-enr 5138 df-nr 5139 df-plr 5140 df-mr 5141 df-ltr 5142 df-0r 5143 df-1r 5144 df-m1r 5145 df-c 5212 df-0 5213 df-1 5214 df-i 5215 df-r 5216 df-plus 5217 df-mul 5218 df-lt 5219 df-sub 5328 df-neg 5330 df-pnf 5459 df-mnf 5460 df-xr 5461 df-ltxr 5462 df-le 5463 df-n 5873