xnn0nnen - Intuitionistic Logic Explorer

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Theorem xnn0nnen 10580

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 5392	. . . . . . . 8 ⊢ ( I ↾ ℕ₀) Fn ℕ₀
2		pnfex 8125	. . . . . . . . 9 ⊢ +∞ ∈ V
3		neg1z 9403	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
4	3	elexi 2783	. . . . . . . . 9 ⊢ -1 ∈ V
5	2, 4	fnsn 5327	. . . . . . . 8 ⊢ {⟨+∞, -1⟩} Fn {+∞}
6	1, 5	pm3.2i 272	. . . . . . 7 ⊢ (( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞})
7		disj 3508	. . . . . . . 8 ⊢ ((ℕ₀ ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ₀ ¬ 𝑥 ∈ {+∞})
8		nn0nepnf 9365	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ≠ +∞)
9		nelsn 3667	. . . . . . . . 9 ⊢ (𝑥 ≠ +∞ → ¬ 𝑥 ∈ {+∞})
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → ¬ 𝑥 ∈ {+∞})
11	7, 10	mprgbir 2563	. . . . . . 7 ⊢ (ℕ₀ ∩ {+∞}) = ∅
12		fnun 5381	. . . . . . 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 3316	. . . . . . 7 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀))
15		df-xnn0 9358	. . . . . . . 8 ⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
16	15	eqcomi 2208	. . . . . . 7 ⊢ (ℕ₀ ∪ {+∞}) = ℕ₀^*
17		fneq12 5366	. . . . . . 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 5327	. . . . . . . . . 10 ⊢ {⟨-1, +∞⟩} Fn {-1}
21	20, 1	pm3.2i 272	. . . . . . . . 9 ⊢ ({⟨-1, +∞⟩} Fn {-1} ∧ ( I ↾ ℕ₀) Fn ℕ₀)
22		disj 3508	. . . . . . . . . 10 ⊢ (({-1} ∩ ℕ₀) = ∅ ↔ ∀𝑥 ∈ {-1} ¬ 𝑥 ∈ ℕ₀)
23		neg1lt0 9143	. . . . . . . . . . . 12 ⊢ -1 < 0
24		nn0nlt0 9320	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℕ₀ → ¬ -1 < 0)
25	23, 24	mt2 641	. . . . . . . . . . 11 ⊢ ¬ -1 ∈ ℕ₀
26		elsni 3650	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {-1} → 𝑥 = -1)
27	26	eleq1d 2273	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {-1} → (𝑥 ∈ ℕ₀ ↔ -1 ∈ ℕ₀))
28	25, 27	mtbiri 676	. . . . . . . . . 10 ⊢ (𝑥 ∈ {-1} → ¬ 𝑥 ∈ ℕ₀)
29	22, 28	mprgbir 2563	. . . . . . . . 9 ⊢ ({-1} ∩ ℕ₀) = ∅
30		fnun 5381	. . . . . . . . 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 5087	. . . . . . . . . 10 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀))
33	2, 4	cnvsn 5164	. . . . . . . . . . 11 ⊢ ^◡{⟨+∞, -1⟩} = {⟨-1, +∞⟩}
34		cnvresid 5347	. . . . . . . . . . 11 ⊢ ^◡( I ↾ ℕ₀) = ( I ↾ ℕ₀)
35	33, 34	uneq12i 3324	. . . . . . . . . 10 ⊢ (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
36	32, 35	eqtri 2225	. . . . . . . . 9 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
37	36	fneq1i 5367	. . . . . . . 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 10332	. . . . . . . . . . 11 ⊢ (-1 ∈ ℤ → (-1..^(-1 + 1)) = {-1})
40	3, 39	ax-mp 5	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = {-1}
41		ax-1cn 8017	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
42	41, 41	negsubdii 8356	. . . . . . . . . . . 12 ⊢ -(1 − 1) = (-1 + 1)
43		1m1e0 9104	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
44	41, 41	subcli 8347	. . . . . . . . . . . . . 14 ⊢ (1 − 1) ∈ ℂ
45		negeq0 8325	. . . . . . . . . . . . . 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 2227	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
49	48	oveq2i 5954	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = (-1..^0)
50	40, 49	eqtr3i 2227	. . . . . . . . 9 ⊢ {-1} = (-1..^0)
51		nn0uz 9682	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
52	50, 51	uneq12i 3324	. . . . . . . 8 ⊢ ({-1} ∪ ℕ₀) = ((-1..^0) ∪ (ℤ_≥‘0))
53	52	fneq2i 5368	. . . . . . 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 9382	. . . . . . . . 9 ⊢ 0 ∈ ℤ
56		neg1rr 9141	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
57		0re 8071	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
58	56, 57, 23	ltleii 8174	. . . . . . . . 9 ⊢ -1 ≤ 0
59		eluz2 9653	. . . . . . . . 9 ⊢ (0 ∈ (ℤ_≥‘-1) ↔ (-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ -1 ≤ 0))
60	3, 55, 58, 59	mpbir3an 1181	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘-1)
61		fzouzsplit 10301	. . . . . . . 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 5368	. . . . . 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 5529	. . . 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 9300	. . . . . 6 ⊢ ℕ₀ ∈ V
69	2	snex 4228	. . . . . 6 ⊢ {+∞} ∈ V
70	68, 69	unex 4487	. . . . 5 ⊢ (ℕ₀ ∪ {+∞}) ∈ V
71	15, 70	eqeltri 2277	. . . 4 ⊢ ℕ₀^* ∈ V
72	71	f1oen 6849	. . 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 10579	. . 3 ⊢ (-1 ∈ ℤ → (ℤ_≥‘-1) ≈ ℕ)
75	3, 74	ax-mp 5	. 2 ⊢ (ℤ_≥‘-1) ≈ ℕ
76	73, 75	entri 6877	1 ⊢ ℕ₀^* ≈ ℕ

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 = wceq 1372 ∈ wcel 2175 ≠ wne 2375 Vcvv 2771 ∪ cun 3163 ∩ cin 3164 ∅c0 3459 {csn 3632 ⟨cop 3635 class class class wbr 4043 I cid 4334 ^◡ccnv 4673 ↾ cres 4676 Fn wfn 5265 –1-1-onto→wf1o 5269 ‘cfv 5270 (class class class)co 5943 ≈ cen 6824 ℂcc 7922 0cc0 7924 1c1 7925 + caddc 7927 +∞cpnf 8103 < clt 8106 ≤ cle 8107 − cmin 8242 -cneg 8243 ℕcn 9035 ℕ₀cn0 9294 ℕ₀^*cxnn0 9357 ℤcz 9371 ℤ_≥cuz 9647 ..^cfzo 10263

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-0id 8032 ax-rnegex 8033 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-er 6619 df-en 6827 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-inn 9036 df-n0 9295 df-xnn0 9358 df-z 9372 df-uz 9648 df-fz 10130 df-fzo 10264

This theorem is referenced by: nninfct 12333

W3C validator