zmin - Metamath Proof Explorer

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Theorem zmin 6221

Description: There is a unique smallest integer greater than or equal to a given real number.

**Assertion**
Ref	Expression
zmin	⊢ (A ∈ ℝ → ∃!x ∈ ℤ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)))

Distinct variable group: x,y,A

**Proof of Theorem zmin**
Step	Hyp	Ref	Expression
1		eqid 1478	. . 3 ⊢ {z ∈ ℤ∣A ≤ z} = {z ∈ ℤ∣A ≤ z}
2	1	uzwo3lem1 6218	. 2 ⊢ (A ∈ ℝ → ∃!x ∈ {z ∈ ℤ∣A ≤ z}∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y)
3		breq2 2628	. . . . . . 7 ⊢ (z = x → (A ≤ z ↔ A ≤ x))
4	3	elrab 1908	. . . . . 6 ⊢ (x ∈ {z ∈ ℤ∣A ≤ z} ↔ (x ∈ ℤ ⋀ A ≤ x))
5		breq2 2628	. . . . . . . . . 10 ⊢ (z = y → (A ≤ z ↔ A ≤ y))
6	5	elrab 1908	. . . . . . . . 9 ⊢ (y ∈ {z ∈ ℤ∣A ≤ z} ↔ (y ∈ ℤ ⋀ A ≤ y))
7	6	imbi1i 186	. . . . . . . 8 ⊢ ((y ∈ {z ∈ ℤ∣A ≤ z} → x ≤ y) ↔ ((y ∈ ℤ ⋀ A ≤ y) → x ≤ y))
8		impexp 347	. . . . . . . 8 ⊢ (((y ∈ ℤ ⋀ A ≤ y) → x ≤ y) ↔ (y ∈ ℤ → (A ≤ y → x ≤ y)))
9	7, 8	bitr 173	. . . . . . 7 ⊢ ((y ∈ {z ∈ ℤ∣A ≤ z} → x ≤ y) ↔ (y ∈ ℤ → (A ≤ y → x ≤ y)))
10	9	ralbii2 1674	. . . . . 6 ⊢ (∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y ↔ ∀y ∈ ℤ (A ≤ y → x ≤ y))
11	4, 10	anbi12i 484	. . . . 5 ⊢ ((x ∈ {z ∈ ℤ∣A ≤ z} ⋀ ∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y) ↔ ((x ∈ ℤ ⋀ A ≤ x) ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)))
12		anass 441	. . . . 5 ⊢ (((x ∈ ℤ ⋀ A ≤ x) ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)) ↔ (x ∈ ℤ ⋀ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y))))
13	11, 12	bitr 173	. . . 4 ⊢ ((x ∈ {z ∈ ℤ∣A ≤ z} ⋀ ∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y) ↔ (x ∈ ℤ ⋀ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y))))
14	13	eubii 1389	. . 3 ⊢ (∃!x(x ∈ {z ∈ ℤ∣A ≤ z} ⋀ ∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y) ↔ ∃!x(x ∈ ℤ ⋀ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y))))
15		df-reu 1654	. . 3 ⊢ (∃!x ∈ {z ∈ ℤ∣A ≤ z}∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y ↔ ∃!x(x ∈ {z ∈ ℤ∣A ≤ z} ⋀ ∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y))
16		df-reu 1654	. . 3 ⊢ (∃!x ∈ ℤ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)) ↔ ∃!x(x ∈ ℤ ⋀ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y))))
17	14, 15, 16	3bitr4r 184	. 2 ⊢ (∃!x ∈ ℤ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)) ↔ ∃!x ∈ {z ∈ ℤ∣A ≤ z}∀y ∈ {z ∈ ℤ∣A ≤ z}x ≤ y)
18	2, 17	sylibr 200	1 ⊢ (A ∈ ℝ → ∃!x ∈ ℤ (A ≤ x ⋀ ∀y ∈ ℤ (A ≤ y → x ≤ y)))

Colors of variables: wff set class

Syntax hints: → wi 3 ⋀ wa 223 ∈ wcel 960 ∃!weu 1382 ∀wral 1648 ∃!wreu 1650 {crab 1651 class class class wbr 2624 ℝcr 5245 ≤ cle 5307 ℤcz 5310

This theorem is referenced by: zmax 6222 zbtwnre 6223

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 964 ax-gen 965 ax-8 966 ax-9 967 ax-10 968 ax-11 969 ax-12 970 ax-13 971 ax-14 972 ax-17 973 ax-4 975 ax-5o 977 ax-6o 980 ax-9o 1125 ax-10o 1142 ax-16 1212 ax-11o 1220 ax-ext 1462 ax-rep 2698 ax-sep 2708 ax-nul 2715 ax-pow 2748 ax-pr 2785 ax-un 2872 ax-inf2 4634

This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 778 df-3an 779 df-ex 983 df-sb 1174 df-eu 1384 df-mo 1385 df-clab 1467 df-cleq 1472 df-clel 1475 df-ne 1590 df-nel 1591 df-ral 1652 df-rex 1653 df-reu 1654 df-rab 1655 df-v 1815 df-sbc 1945 df-csb 2005 df-dif 2052 df-un 2053 df-in 2054 df-ss 2056 df-pss 2058 df-nul 2284 df-if 2366 df-pw 2406 df-sn 2416 df-pr 2417 df-tp 2419 df-op 2420 df-uni 2508 df-int 2538 df-iun 2572 df-br 2625 df-opab 2672 df-tr 2686 df-eprel 2838 df-id 2841 df-po 2846 df-so 2856 df-fr 2923 df-we 2940 df-ord 2957 df-on 2958 df-lim 2959 df-suc 2960 df-om 3138 df-xp 3190 df-rel 3191 df-cnv 3192 df-co 3193 df-dm 3194 df-rn 3195 df-res 3196 df-ima 3197 df-fun 3198 df-fn 3199 df-f 3200 df-f1 3201 df-fo 3202 df-f1o 3203 df-fv 3204 df-rdg 3938 df-opr 3971 df-oprab 3972 df-1st 4085 df-2nd 4086 df-1o 4139 df-oadd 4141 df-omul 4142 df-er 4267 df-ec 4269 df-qs 4272 df-en 4374 df-dom 4375 df-sdom 4376 df-ni 5012 df-pli 5013 df-mi 5014 df-lti 5015 df-plpq 5047 df-mpq 5048 df-enq 5049 df-nq 5050 df-plq 5051 df-mq 5052 df-rq 5053 df-ltq 5054 df-1q 5055 df-np 5098 df-1p 5099 df-plp 5100 df-mp 5101 df-ltp 5102 df-plpr 5176 df-mpr 5177 df-enr 5178 df-nr 5179 df-plr 5180 df-mr 5181 df-ltr 5182 df-0r 5183 df-1r 5184 df-m1r 5185 df-c 5252 df-0 5253 df-1 5254 df-i 5255 df-r 5256 df-plus 5257 df-mul 5258 df-lt 5259 df-sub 5368 df-neg 5370 df-pnf 5499 df-mnf 5500 df-xr 5501 df-ltxr 5502 df-le 5503 df-n 5927 df-n0 6102 df-z 6138