xnn0nnen - Intuitionistic Logic Explorer

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Theorem xnn0nnen 10799

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 5476	. . . . . . . 8 ⊢ ( I ↾ ℕ₀) Fn ℕ₀
2		pnfex 8327	. . . . . . . . 9 ⊢ +∞ ∈ V
3		neg1z 9609	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
4	3	elexi 2826	. . . . . . . . 9 ⊢ -1 ∈ V
5	2, 4	fnsn 5410	. . . . . . . 8 ⊢ {⟨+∞, -1⟩} Fn {+∞}
6	1, 5	pm3.2i 272	. . . . . . 7 ⊢ (( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞})
7		disj 3557	. . . . . . . 8 ⊢ ((ℕ₀ ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ₀ ¬ 𝑥 ∈ {+∞})
8		nn0nepnf 9571	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ≠ +∞)
9		nelsn 3724	. . . . . . . . 9 ⊢ (𝑥 ≠ +∞ → ¬ 𝑥 ∈ {+∞})
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → ¬ 𝑥 ∈ {+∞})
11	7, 10	mprgbir 2600	. . . . . . 7 ⊢ (ℕ₀ ∩ {+∞}) = ∅
12		fnun 5464	. . . . . . 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 3363	. . . . . . 7 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀))
15		df-xnn0 9564	. . . . . . . 8 ⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
16	15	eqcomi 2236	. . . . . . 7 ⊢ (ℕ₀ ∪ {+∞}) = ℕ₀^*
17		fneq12 5449	. . . . . . 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 5410	. . . . . . . . . 10 ⊢ {⟨-1, +∞⟩} Fn {-1}
21	20, 1	pm3.2i 272	. . . . . . . . 9 ⊢ ({⟨-1, +∞⟩} Fn {-1} ∧ ( I ↾ ℕ₀) Fn ℕ₀)
22		disj 3557	. . . . . . . . . 10 ⊢ (({-1} ∩ ℕ₀) = ∅ ↔ ∀𝑥 ∈ {-1} ¬ 𝑥 ∈ ℕ₀)
23		neg1lt0 9345	. . . . . . . . . . . 12 ⊢ -1 < 0
24		nn0nlt0 9522	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℕ₀ → ¬ -1 < 0)
25	23, 24	mt2 645	. . . . . . . . . . 11 ⊢ ¬ -1 ∈ ℕ₀
26		elsni 3707	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {-1} → 𝑥 = -1)
27	26	eleq1d 2301	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {-1} → (𝑥 ∈ ℕ₀ ↔ -1 ∈ ℕ₀))
28	25, 27	mtbiri 682	. . . . . . . . . 10 ⊢ (𝑥 ∈ {-1} → ¬ 𝑥 ∈ ℕ₀)
29	22, 28	mprgbir 2600	. . . . . . . . 9 ⊢ ({-1} ∩ ℕ₀) = ∅
30		fnun 5464	. . . . . . . . 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 5168	. . . . . . . . . 10 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀))
33	2, 4	cnvsn 5245	. . . . . . . . . . 11 ⊢ ^◡{⟨+∞, -1⟩} = {⟨-1, +∞⟩}
34		cnvresid 5430	. . . . . . . . . . 11 ⊢ ^◡( I ↾ ℕ₀) = ( I ↾ ℕ₀)
35	33, 34	uneq12i 3371	. . . . . . . . . 10 ⊢ (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
36	32, 35	eqtri 2253	. . . . . . . . 9 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
37	36	fneq1i 5450	. . . . . . . 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 10550	. . . . . . . . . . 11 ⊢ (-1 ∈ ℤ → (-1..^(-1 + 1)) = {-1})
40	3, 39	ax-mp 5	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = {-1}
41		ax-1cn 8220	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
42	41, 41	negsubdii 8558	. . . . . . . . . . . 12 ⊢ -(1 − 1) = (-1 + 1)
43		1m1e0 9306	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
44	41, 41	subcli 8549	. . . . . . . . . . . . . 14 ⊢ (1 − 1) ∈ ℂ
45		negeq0 8527	. . . . . . . . . . . . . 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 2255	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
49	48	oveq2i 6061	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = (-1..^0)
50	40, 49	eqtr3i 2255	. . . . . . . . 9 ⊢ {-1} = (-1..^0)
51		nn0uz 9889	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
52	50, 51	uneq12i 3371	. . . . . . . 8 ⊢ ({-1} ∪ ℕ₀) = ((-1..^0) ∪ (ℤ_≥‘0))
53	52	fneq2i 5451	. . . . . . 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 9588	. . . . . . . . 9 ⊢ 0 ∈ ℤ
56		neg1rr 9343	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
57		0re 8274	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
58	56, 57, 23	ltleii 8376	. . . . . . . . 9 ⊢ -1 ≤ 0
59		eluz2 9859	. . . . . . . . 9 ⊢ (0 ∈ (ℤ_≥‘-1) ↔ (-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ -1 ≤ 0))
60	3, 55, 58, 59	mpbir3an 1206	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘-1)
61		fzouzsplit 10515	. . . . . . . 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 5451	. . . . . 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 5622	. . . 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 9502	. . . . . 6 ⊢ ℕ₀ ∈ V
69	2	snex 4298	. . . . . 6 ⊢ {+∞} ∈ V
70	68, 69	unex 4562	. . . . 5 ⊢ (ℕ₀ ∪ {+∞}) ∈ V
71	15, 70	eqeltri 2305	. . . 4 ⊢ ℕ₀^* ∈ V
72	71	f1oen 6998	. . 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 10798	. . 3 ⊢ (-1 ∈ ℤ → (ℤ_≥‘-1) ≈ ℕ)
75	3, 74	ax-mp 5	. 2 ⊢ (ℤ_≥‘-1) ≈ ℕ
76	73, 75	entri 7026	1 ⊢ ℕ₀^* ≈ ℕ

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 Vcvv 2813 ∪ cun 3209 ∩ cin 3210 ∅c0 3508 {csn 3689 ⟨cop 3692 class class class wbr 4109 I cid 4409 ^◡ccnv 4748 ↾ cres 4751 Fn wfn 5347 –1-1-onto→wf1o 5351 ‘cfv 5352 (class class class)co 6050 ≈ cen 6973 ℂcc 8125 0cc0 8127 1c1 8128 + caddc 8130 +∞cpnf 8305 < clt 8308 ≤ cle 8309 − cmin 8444 -cneg 8445 ℕcn 9237 ℕ₀cn0 9496 ℕ₀^*cxnn0 9563 ℤcz 9577 ℤ_≥cuz 9853 ..^cfzo 10476

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-addass 8229 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-er 6767 df-en 6976 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-n0 9497 df-xnn0 9564 df-z 9578 df-uz 9854 df-fz 10343 df-fzo 10477

This theorem is referenced by: nninfct 12737

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