xnn0nnen - Intuitionistic Logic Explorer

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Theorem xnn0nnen 10895

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 5504	. . . . . . . 8 ⊢ ( I ↾ ℕ₀) Fn ℕ₀
2		pnfex 8385	. . . . . . . . 9 ⊢ +∞ ∈ V
3		neg1z 9687	. . . . . . . . . 10 ⊢ -1 ∈ ℤ
4	3	elexi 2834	. . . . . . . . 9 ⊢ -1 ∈ V
5	2, 4	fnsn 5438	. . . . . . . 8 ⊢ {⟨+∞, -1⟩} Fn {+∞}
6	1, 5	pm3.2i 272	. . . . . . 7 ⊢ (( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞})
7		disj 3573	. . . . . . . 8 ⊢ ((ℕ₀ ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ₀ ¬ 𝑥 ∈ {+∞})
8		nn0nepnf 9649	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ≠ +∞)
9		nelsn 3744	. . . . . . . . 9 ⊢ (𝑥 ≠ +∞ → ¬ 𝑥 ∈ {+∞})
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → ¬ 𝑥 ∈ {+∞})
11	7, 10	mprgbir 2608	. . . . . . 7 ⊢ (ℕ₀ ∩ {+∞}) = ∅
12		fnun 5492	. . . . . . 7 ⊢ (((( I ↾ ℕ₀) Fn ℕ₀ ∧ {⟨+∞, -1⟩} Fn {+∞}) ∧ (ℕ₀ ∩ {+∞}) = ∅) → (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞}))
13	6, 11, 12	mp2an 430	. . . . . 6 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞})
14		uncom 3373	. . . . . . 7 ⊢ (( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀))
15		df-xnn0 9642	. . . . . . . 8 ⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
16	15	eqcomi 2242	. . . . . . 7 ⊢ (ℕ₀ ∪ {+∞}) = ℕ₀^*
17		fneq12 5477	. . . . . . 7 ⊢ (((( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) = ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) ∧ (ℕ₀ ∪ {+∞}) = ℕ₀^) → ((( I ↾ ℕ₀) ∪ {⟨+∞, -1⟩}) Fn (ℕ₀ ∪ {+∞}) ↔ ({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) Fn ℕ₀^))
18	14, 16, 17	mp2an 430	. . . . . 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 5438	. . . . . . . . . 10 ⊢ {⟨ -1, +∞⟩} Fn { -1}
21	20, 1	pm3.2i 272	. . . . . . . . 9 ⊢ ({⟨ -1, +∞⟩} Fn { -1} ∧ ( I ↾ ℕ₀) Fn ℕ₀)
22		disj 3573	. . . . . . . . . 10 ⊢ (({ -1} ∩ ℕ₀) = ∅ ↔ ∀𝑥 ∈ { -1} ¬ 𝑥 ∈ ℕ₀)
23		neg1lt0 9421	. . . . . . . . . . . 12 ⊢ -1 < 0
24		nn0nlt0 9600	. . . . . . . . . . . 12 ⊢ ( -1 ∈ ℕ₀ → ¬ -1 < 0)
25	23, 24	mt2 649	. . . . . . . . . . 11 ⊢ ¬ -1 ∈ ℕ₀
26		elsni 3727	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ { -1} → 𝑥 = -1)
27	26	eleq1d 2307	. . . . . . . . . . 11 ⊢ (𝑥 ∈ { -1} → (𝑥 ∈ ℕ₀ ↔ -1 ∈ ℕ₀))
28	25, 27	mtbiri 686	. . . . . . . . . 10 ⊢ (𝑥 ∈ { -1} → ¬ 𝑥 ∈ ℕ₀)
29	22, 28	mprgbir 2608	. . . . . . . . 9 ⊢ ({ -1} ∩ ℕ₀) = ∅
30		fnun 5492	. . . . . . . . 9 ⊢ ((({⟨ -1, +∞⟩} Fn { -1} ∧ ( I ↾ ℕ₀) Fn ℕ₀) ∧ ({ -1} ∩ ℕ₀) = ∅) → ({⟨ -1, +∞⟩} ∪ ( I ↾ ℕ₀)) Fn ({ -1} ∪ ℕ₀))
31	21, 29, 30	mp2an 430	. . . . . . . 8 ⊢ ({⟨ -1, +∞⟩} ∪ ( I ↾ ℕ₀)) Fn ({ -1} ∪ ℕ₀)
32		cnvun 5196	. . . . . . . . . 10 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀))
33	2, 4	cnvsn 5273	. . . . . . . . . . 11 ⊢ ^◡{⟨+∞, -1⟩} = {⟨ -1, +∞⟩}
34		cnvresid 5458	. . . . . . . . . . 11 ⊢ ^◡( I ↾ ℕ₀) = ( I ↾ ℕ₀)
35	33, 34	uneq12i 3381	. . . . . . . . . 10 ⊢ (^◡{⟨+∞, -1⟩} ∪ ^◡( I ↾ ℕ₀)) = ({⟨ -1, +∞⟩} ∪ ( I ↾ ℕ₀))
36	32, 35	eqtri 2259	. . . . . . . . 9 ⊢ ^◡({⟨+∞, -1⟩} ∪ ( I ↾ ℕ₀)) = ({⟨ -1, +∞⟩} ∪ ( I ↾ ℕ₀))
37	36	fneq1i 5478	. . . . . . . 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 10640	. . . . . . . . . . 11 ⊢ ( -1 ∈ ℤ → ( -1..^( -1 + 1)) = { -1})
40	3, 39	ax-mp 5	. . . . . . . . . 10 ⊢ ( -1..^( -1 + 1)) = { -1}
41		ax-1cn 8278	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
42	41, 41	negsubdii 8618	. . . . . . . . . . . 12 ⊢ -(1 − 1) = ( -1 + 1)
43		1m1e0 9382	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
44	41, 41	subcli 8609	. . . . . . . . . . . . . 14 ⊢ (1 − 1) ∈ ℂ
45		negeq0 8587	. . . . . . . . . . . . . 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 2261	. . . . . . . . . . 11 ⊢ ( -1 + 1) = 0
49	48	oveq2i 6099	. . . . . . . . . 10 ⊢ ( -1..^( -1 + 1)) = ( -1..^0)
50	40, 49	eqtr3i 2261	. . . . . . . . 9 ⊢ { -1} = ( -1..^0)
51		nn0uz 9973	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
52	50, 51	uneq12i 3381	. . . . . . . 8 ⊢ ({ -1} ∪ ℕ₀) = (( -1..^0) ∪ (ℤ_≥‘0))
53	52	fneq2i 5479	. . . . . . 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 9666	. . . . . . . . 9 ⊢ 0 ∈ ℤ
56		neg1rr 9419	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
57		0re 8332	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
58	56, 57, 23	ltleii 8435	. . . . . . . . 9 ⊢ -1 ≤ 0
59		eluz2 9943	. . . . . . . . 9 ⊢ (0 ∈ (ℤ_≥‘ -1) ↔ ( -1 ∈ ℤ ∧ 0 ∈ ℤ ∧ -1 ≤ 0))
60	3, 55, 58, 59	mpbir3an 1210	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘ -1)
61		fzouzsplit 10605	. . . . . . . 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 5479	. . . . . 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 5650	. . . 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 9580	. . . . . 6 ⊢ ℕ₀ ∈ V
69	2	snex 4324	. . . . . 6 ⊢ {+∞} ∈ V
70	68, 69	unex 4589	. . . . 5 ⊢ (ℕ₀ ∪ {+∞}) ∈ V
71	15, 70	eqeltri 2311	. . . 4 ⊢ ℕ₀^* ∈ V
72	71	f1oen 7049	. . 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 10894	. . 3 ⊢ ( -1 ∈ ℤ → (ℤ_≥‘ -1) ≈ ℕ)
75	3, 74	ax-mp 5	. 2 ⊢ (ℤ_≥‘ -1) ≈ ℕ
76	73, 75	entri 7077	1 ⊢ ℕ₀^* ≈ ℕ

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 Vcvv 2821 ∪ cun 3218 ∩ cin 3219 ∅c0 3520 {csn 3709 ⟨cop 3712 class class class wbr 4131 I cid 4435 ^◡ccnv 4775 ↾ cres 4778 Fn wfn 5375 –1-1-onto→wf1o 5379 ‘cfv 5380 (class class class)co 6088 ≈ cen 7024 ℂcc 8183 0cc0 8185 1c1 8186 + caddc 8188 +∞cpnf 8363 < clt 8366 ≤ cle 8367 − cmin 8504 -cneg 8505 ℕcn 9313 ℕ₀cn0 9574 ℕ₀^*cxnn0 9641 ℤcz 9655 ℤ_≥cuz 9937 ..^cfzo 10566

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-er 6811 df-en 7027 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-inn 9314 df-n0 9575 df-xnn0 9642 df-z 9656 df-uz 9938 df-fz 10429 df-fzo 10567

This theorem is used by: nninfct 12843

W3C validator