xnn0nnen - Intuitionistic Logic Explorer

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Theorem xnn0nnen 10511

Description: The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)

**Assertion**
Ref	Expression
xnn0nnen	⊢ ℕ₀^* ≈ ℕ

**Proof of Theorem xnn0nnen**
Step	Hyp	Ref	Expression
1		fnresi 5372	. . . . . . . 8 ⊢ ( I ↾ ℕ₀) Fn ℕ₀
2		pnfex 8075	. . . . . . . . 9 ⊢ +∞ ∈ V
3		neg1z 9352	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
4	3	elexi 2772	. . . . . . . . 9 ⊢ -1 ∈ V
5	2, 4	fnsn 5309	. . . . . . . 8 ⊢ {⟨+∞, -1⟩} Fn {+∞}
6	1, 5	pm3.2i 272	. . . . . . 7 ⊢ (( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞})
7		disj 3496	. . . . . . . 8 ⊢ ((ℕ₀ ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ₀ ¬ 𝑥 ∈ {+∞})
8		nn0nepnf 9314	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ≠ +∞)
9		nelsn 3654	. . . . . . . . 9 ⊢ (𝑥 ≠ +∞ → ¬ 𝑥 ∈ {+∞})
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → ¬ 𝑥 ∈ {+∞})
11	7, 10	mprgbir 2552	. . . . . . 7 ⊢ (ℕ₀ ∩ {+∞}) = ∅
12		fnun 5361	. . . . . . 7 ⊢ (((( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞}) ∧ (ℕ₀ ∩ {+∞}) = ∅) → (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞}))
13	6, 11, 12	mp2an 426	. . . . . 6 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞})
14		uncom 3304	. . . . . . 7 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀))
15		df-xnn0 9307	. . . . . . . 8 ⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
16	15	eqcomi 2197	. . . . . . 7 ⊢ (ℕ₀ ∪ {+∞}) = ℕ₀^*
17		fneq12 5348	. . . . . . 7 ⊢ (((( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) ∧ (ℕ₀ ∪ {+∞}) = ℕ₀^) → ((( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞}) ↔ ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^))
18	14, 16, 17	mp2an 426	. . . . . 6 ⊢ ((( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞}) ↔ ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^*)
19	13, 18	mpbi 145	. . . . 5 ⊢ ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^*
20	4, 2	fnsn 5309	. . . . . . . . . 10 ⊢ {⟨-1, +∞⟩} Fn {-1}
21	20, 1	pm3.2i 272	. . . . . . . . 9 ⊢ ({⟨-1, +∞⟩} Fn {-1} ∧ ( I ↾ ℕ₀) Fn ℕ₀)
22		disj 3496	. . . . . . . . . 10 ⊢ (({-1} ∩ ℕ₀) = ∅ ↔ ∀𝑥 ∈ {-1} ¬ 𝑥 ∈ ℕ₀)
23		neg1lt0 9092	. . . . . . . . . . . 12 ⊢ -1 < 0
24		nn0nlt0 9269	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℕ₀ → ¬ -1 < 0)
25	23, 24	mt2 641	. . . . . . . . . . 11 ⊢ ¬ -1 ∈ ℕ₀
26		elsni 3637	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {-1} → 𝑥 = -1)
27	26	eleq1d 2262	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {-1} → (𝑥 ∈ ℕ₀ ↔ -1 ∈ ℕ₀))
28	25, 27	mtbiri 676	. . . . . . . . . 10 ⊢ (𝑥 ∈ {-1} → ¬ 𝑥 ∈ ℕ₀)
29	22, 28	mprgbir 2552	. . . . . . . . 9 ⊢ ({-1} ∩ ℕ₀) = ∅
30		fnun 5361	. . . . . . . . 9 ⊢ ((({⟨-1, +∞⟩} Fn {-1} ∧ ( I ↾ ℕ₀) Fn ℕ₀) ∧ ({-1} ∩ ℕ₀) = ∅) → ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀))
31	21, 29, 30	mp2an 426	. . . . . . . 8 ⊢ ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀)
32		cnvun 5072	. . . . . . . . . 10 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀))
33	2, 4	cnvsn 5149	. . . . . . . . . . 11 ⊢ ^◡{⟨+∞, -1⟩} = {⟨-1, +∞⟩}
34		cnvresid 5329	. . . . . . . . . . 11 ⊢ ^◡( I ↾ ℕ₀) = ( I ↾ ℕ₀)
35	33, 34	uneq12i 3312	. . . . . . . . . 10 ⊢ (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
36	32, 35	eqtri 2214	. . . . . . . . 9 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
37	36	fneq1i 5349	. . . . . . . 8 ⊢ (^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀) ↔ ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀))
38	31, 37	mpbir 146	. . . . . . 7 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀)
39		fzosn 10275	. . . . . . . . . . 11 ⊢ (-1 ∈ ℤ → (-1..^(-1 + 1)) = {-1})
40	3, 39	ax-mp 5	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = {-1}
41		ax-1cn 7967	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
42	41, 41	negsubdii 8306	. . . . . . . . . . . 12 ⊢ -(1 − 1) = (-1 + 1)
43		1m1e0 9053	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
44	41, 41	subcli 8297	. . . . . . . . . . . . . 14 ⊢ (1 − 1) ∈ ℂ
45		negeq0 8275	. . . . . . . . . . . . . 14 ⊢ ((1 − 1) ∈ ℂ → ((1 − 1) = 0 ↔ -(1 − 1) = 0))
46	44, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((1 − 1) = 0 ↔ -(1 − 1) = 0)
47	43, 46	mpbi 145	. . . . . . . . . . . 12 ⊢ -(1 − 1) = 0
48	42, 47	eqtr3i 2216	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
49	48	oveq2i 5930	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = (-1..^0)
50	40, 49	eqtr3i 2216	. . . . . . . . 9 ⊢ {-1} = (-1..^0)
51		nn0uz 9630	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
52	50, 51	uneq12i 3312	. . . . . . . 8 ⊢ ({-1} ∪ ℕ₀) = ((-1..^0) ∪ (ℤ_≥‘0))
53	52	fneq2i 5350	. . . . . . 7 ⊢ (^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ({-1} ∪ ℕ₀) ↔ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ((-1..^0) ∪ (ℤ_≥‘0)))
54	38, 53	mpbi 145	. . . . . 6 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ((-1..^0) ∪ (ℤ_≥‘0))
55		0z 9331	. . . . . . . . 9 ⊢ 0 ∈ ℤ
56		neg1rr 9090	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
57		0re 8021	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
58	56, 57, 23	ltleii 8124	. . . . . . . . 9 ⊢ -1 ≤ 0
59		eluz2 9601	. . . . . . . . 9 ⊢ (0 ∈ (ℤ_≥‘-1) ↔ (-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ -1 ≤ 0))
60	3, 55, 58, 59	mpbir3an 1181	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘-1)
61		fzouzsplit 10249	. . . . . . . 8 ⊢ (0 ∈ (ℤ_≥‘-1) → (ℤ_≥‘-1) = ((-1..^0) ∪ (ℤ_≥‘0)))
62	60, 61	ax-mp 5	. . . . . . 7 ⊢ (ℤ_≥‘-1) = ((-1..^0) ∪ (ℤ_≥‘0))
63	62	fneq2i 5350	. . . . . 6 ⊢ (^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn (ℤ_≥‘-1) ↔ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ((-1..^0) ∪ (ℤ_≥‘0)))
64	54, 63	mpbir 146	. . . . 5 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn (ℤ_≥‘-1)
65	19, 64	pm3.2i 272	. . . 4 ⊢ (({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^* ∧ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn (ℤ_≥‘-1))
66		dff1o4 5509	. . . 4 ⊢ (({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)):ℕ₀^–1-1-onto→(ℤ_≥‘-1) ↔ (({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^ ∧ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn (ℤ_≥‘-1)))
67	65, 66	mpbir 146	. . 3 ⊢ ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)):ℕ₀^*–1-1-onto→(ℤ_≥‘-1)
68		nn0ex 9249	. . . . . 6 ⊢ ℕ₀ ∈ V
69	2	snex 4215	. . . . . 6 ⊢ {+∞} ∈ V
70	68, 69	unex 4473	. . . . 5 ⊢ (ℕ₀ ∪ {+∞}) ∈ V
71	15, 70	eqeltri 2266	. . . 4 ⊢ ℕ₀^* ∈ V
72	71	f1oen 6815	. . 3 ⊢ (({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)):ℕ₀^–1-1-onto→(ℤ_≥‘-1) → ℕ₀^ ≈ (ℤ_≥‘-1))
73	67, 72	ax-mp 5	. 2 ⊢ ℕ₀^* ≈ (ℤ_≥‘-1)
74		uzennn 10510	. . 3 ⊢ (-1 ∈ ℤ → (ℤ_≥‘-1) ≈ ℕ)
75	3, 74	ax-mp 5	. 2 ⊢ (ℤ_≥‘-1) ≈ ℕ
76	73, 75	entri 6842	1 ⊢ ℕ₀^* ≈ ℕ

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 Vcvv 2760 ∪ cun 3152 ∩ cin 3153 ∅c0 3447 {csn 3619 ⟨cop 3622 class class class wbr 4030 I cid 4320 ^◡ccnv 4659 ↾ cres 4662 Fn wfn 5250 –1-1-onto→wf1o 5254 ‘cfv 5255 (class class class)co 5919 ≈ cen 6794 ℂcc 7872 0cc0 7874 1c1 7875 + caddc 7877 +∞cpnf 8053 < clt 8056 ≤ cle 8057 − cmin 8192 -cneg 8193 ℕcn 8984 ℕ₀cn0 9243 ℕ₀^*cxnn0 9306 ℤcz 9320 ℤ_≥cuz 9595 ..^cfzo 10211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-er 6589 df-en 6797 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-inn 8985 df-n0 9244 df-xnn0 9307 df-z 9321 df-uz 9596 df-fz 10078 df-fzo 10212

This theorem is referenced by: nninfct 12181

W3C validator