xnn0nnen - Intuitionistic Logic Explorer

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Theorem xnn0nnen 10654

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 5440	. . . . . . . 8 ⊢ ( I ↾ ℕ₀) Fn ℕ₀
2		pnfex 8196	. . . . . . . . 9 ⊢ +∞ ∈ V
3		neg1z 9474	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
4	3	elexi 2812	. . . . . . . . 9 ⊢ -1 ∈ V
5	2, 4	fnsn 5374	. . . . . . . 8 ⊢ {⟨+∞, -1⟩} Fn {+∞}
6	1, 5	pm3.2i 272	. . . . . . 7 ⊢ (( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞})
7		disj 3540	. . . . . . . 8 ⊢ ((ℕ₀ ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ₀ ¬ 𝑥 ∈ {+∞})
8		nn0nepnf 9436	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ≠ +∞)
9		nelsn 3701	. . . . . . . . 9 ⊢ (𝑥 ≠ +∞ → ¬ 𝑥 ∈ {+∞})
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → ¬ 𝑥 ∈ {+∞})
11	7, 10	mprgbir 2588	. . . . . . 7 ⊢ (ℕ₀ ∩ {+∞}) = ∅
12		fnun 5428	. . . . . . 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 3348	. . . . . . 7 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀))
15		df-xnn0 9429	. . . . . . . 8 ⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
16	15	eqcomi 2233	. . . . . . 7 ⊢ (ℕ₀ ∪ {+∞}) = ℕ₀^*
17		fneq12 5413	. . . . . . 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 5374	. . . . . . . . . 10 ⊢ {⟨-1, +∞⟩} Fn {-1}
21	20, 1	pm3.2i 272	. . . . . . . . 9 ⊢ ({⟨-1, +∞⟩} Fn {-1} ∧ ( I ↾ ℕ₀) Fn ℕ₀)
22		disj 3540	. . . . . . . . . 10 ⊢ (({-1} ∩ ℕ₀) = ∅ ↔ ∀𝑥 ∈ {-1} ¬ 𝑥 ∈ ℕ₀)
23		neg1lt0 9214	. . . . . . . . . . . 12 ⊢ -1 < 0
24		nn0nlt0 9391	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℕ₀ → ¬ -1 < 0)
25	23, 24	mt2 643	. . . . . . . . . . 11 ⊢ ¬ -1 ∈ ℕ₀
26		elsni 3684	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {-1} → 𝑥 = -1)
27	26	eleq1d 2298	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {-1} → (𝑥 ∈ ℕ₀ ↔ -1 ∈ ℕ₀))
28	25, 27	mtbiri 679	. . . . . . . . . 10 ⊢ (𝑥 ∈ {-1} → ¬ 𝑥 ∈ ℕ₀)
29	22, 28	mprgbir 2588	. . . . . . . . 9 ⊢ ({-1} ∩ ℕ₀) = ∅
30		fnun 5428	. . . . . . . . 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 5133	. . . . . . . . . 10 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀))
33	2, 4	cnvsn 5210	. . . . . . . . . . 11 ⊢ ^◡{⟨+∞, -1⟩} = {⟨-1, +∞⟩}
34		cnvresid 5394	. . . . . . . . . . 11 ⊢ ^◡( I ↾ ℕ₀) = ( I ↾ ℕ₀)
35	33, 34	uneq12i 3356	. . . . . . . . . 10 ⊢ (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
36	32, 35	eqtri 2250	. . . . . . . . 9 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = ({⟨-1, +∞⟩} ∪ ( I ↾ ℕ₀))
37	36	fneq1i 5414	. . . . . . . 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 10406	. . . . . . . . . . 11 ⊢ (-1 ∈ ℤ → (-1..^(-1 + 1)) = {-1})
40	3, 39	ax-mp 5	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = {-1}
41		ax-1cn 8088	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
42	41, 41	negsubdii 8427	. . . . . . . . . . . 12 ⊢ -(1 − 1) = (-1 + 1)
43		1m1e0 9175	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
44	41, 41	subcli 8418	. . . . . . . . . . . . . 14 ⊢ (1 − 1) ∈ ℂ
45		negeq0 8396	. . . . . . . . . . . . . 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 2252	. . . . . . . . . . 11 ⊢ (-1 + 1) = 0
49	48	oveq2i 6011	. . . . . . . . . 10 ⊢ (-1..^(-1 + 1)) = (-1..^0)
50	40, 49	eqtr3i 2252	. . . . . . . . 9 ⊢ {-1} = (-1..^0)
51		nn0uz 9753	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
52	50, 51	uneq12i 3356	. . . . . . . 8 ⊢ ({-1} ∪ ℕ₀) = ((-1..^0) ∪ (ℤ_≥‘0))
53	52	fneq2i 5415	. . . . . . 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 9453	. . . . . . . . 9 ⊢ 0 ∈ ℤ
56		neg1rr 9212	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
57		0re 8142	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
58	56, 57, 23	ltleii 8245	. . . . . . . . 9 ⊢ -1 ≤ 0
59		eluz2 9724	. . . . . . . . 9 ⊢ (0 ∈ (ℤ_≥‘-1) ↔ (-1 ∈ ℤ ∧ 0 ∈ ℤ ∧ -1 ≤ 0))
60	3, 55, 58, 59	mpbir3an 1203	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘-1)
61		fzouzsplit 10373	. . . . . . . 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 5415	. . . . . 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 5579	. . . 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 9371	. . . . . 6 ⊢ ℕ₀ ∈ V
69	2	snex 4268	. . . . . 6 ⊢ {+∞} ∈ V
70	68, 69	unex 4531	. . . . 5 ⊢ (ℕ₀ ∪ {+∞}) ∈ V
71	15, 70	eqeltri 2302	. . . 4 ⊢ ℕ₀^* ∈ V
72	71	f1oen 6908	. . 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 10653	. . 3 ⊢ (-1 ∈ ℤ → (ℤ_≥‘-1) ≈ ℕ)
75	3, 74	ax-mp 5	. 2 ⊢ (ℤ_≥‘-1) ≈ ℕ
76	73, 75	entri 6936	1 ⊢ ℕ₀^* ≈ ℕ

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 Vcvv 2799 ∪ cun 3195 ∩ cin 3196 ∅c0 3491 {csn 3666 ⟨cop 3669 class class class wbr 4082 I cid 4378 ^◡ccnv 4717 ↾ cres 4720 Fn wfn 5312 –1-1-onto→wf1o 5316 ‘cfv 5317 (class class class)co 6000 ≈ cen 6883 ℂcc 7993 0cc0 7995 1c1 7996 + caddc 7998 +∞cpnf 8174 < clt 8177 ≤ cle 8178 − cmin 8313 -cneg 8314 ℕcn 9106 ℕ₀cn0 9365 ℕ₀^*cxnn0 9428 ℤcz 9442 ℤ_≥cuz 9718 ..^cfzo 10334

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-addcom 8095 ax-addass 8097 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-0id 8103 ax-rnegex 8104 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-er 6678 df-en 6886 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-inn 9107 df-n0 9366 df-xnn0 9429 df-z 9443 df-uz 9719 df-fz 10201 df-fzo 10335

This theorem is referenced by: nninfct 12557

W3C validator