Theorem nn0suc 4223

Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
nn0suc	⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∃x ∈ 𝜔 A = suc x))

Distinct variable group:   x,A

Proof of Theorem nn0suc

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2028	. . 3 ⊢ (y = ∅ → (y = ∅ ↔ ∅ = ∅))
2		eqeq1 2028	. . . 4 ⊢ (y = ∅ → (y = suc x ↔ ∅ = suc x))
3	2	rexbidv 2303	. . 3 ⊢ (y = ∅ → (∃x ∈ 𝜔 y = suc x ↔ ∃x ∈ 𝜔 ∅ = suc x))
4	1, 3	orbi12d 695	. 2 ⊢ (y = ∅ → ((y = ∅ ∨ ∃x ∈ 𝜔 y = suc x) ↔ (∅ = ∅ ∨ ∃x ∈ 𝜔 ∅ = suc x)))
5		eqeq1 2028	. . 3 ⊢ (y = z → (y = ∅ ↔ z = ∅))
6		eqeq1 2028	. . . 4 ⊢ (y = z → (y = suc x ↔ z = suc x))
7	6	rexbidv 2303	. . 3 ⊢ (y = z → (∃x ∈ 𝜔 y = suc x ↔ ∃x ∈ 𝜔 z = suc x))
8	5, 7	orbi12d 695	. 2 ⊢ (y = z → ((y = ∅ ∨ ∃x ∈ 𝜔 y = suc x) ↔ (z = ∅ ∨ ∃x ∈ 𝜔 z = suc x)))
9		eqeq1 2028	. . 3 ⊢ (y = suc z → (y = ∅ ↔ suc z = ∅))
10		eqeq1 2028	. . . 4 ⊢ (y = suc z → (y = suc x ↔ suc z = suc x))
11	10	rexbidv 2303	. . 3 ⊢ (y = suc z → (∃x ∈ 𝜔 y = suc x ↔ ∃x ∈ 𝜔 suc z = suc x))
12	9, 11	orbi12d 695	. 2 ⊢ (y = suc z → ((y = ∅ ∨ ∃x ∈ 𝜔 y = suc x) ↔ (suc z = ∅ ∨ ∃x ∈ 𝜔 suc z = suc x)))
13		eqeq1 2028	. . 3 ⊢ (y = A → (y = ∅ ↔ A = ∅))
14		eqeq1 2028	. . . 4 ⊢ (y = A → (y = suc x ↔ A = suc x))
15	14	rexbidv 2303	. . 3 ⊢ (y = A → (∃x ∈ 𝜔 y = suc x ↔ ∃x ∈ 𝜔 A = suc x))
16	13, 15	orbi12d 695	. 2 ⊢ (y = A → ((y = ∅ ∨ ∃x ∈ 𝜔 y = suc x) ↔ (A = ∅ ∨ ∃x ∈ 𝜔 A = suc x)))
17		eqid 2022	. . 3 ⊢ ∅ = ∅
18	17	orci 637	. 2 ⊢ (∅ = ∅ ∨ ∃x ∈ 𝜔 ∅ = suc x)
19		eqid 2022	. . . . 5 ⊢ suc z = suc z
20		suceq 4062	. . . . . . 7 ⊢ (x = z → suc x = suc z)
21	20	eqeq2d 2033	. . . . . 6 ⊢ (x = z → (suc z = suc x ↔ suc z = suc z))
22	21	rspcev 2631	. . . . 5 ⊢ ((z ∈ 𝜔 ∧ suc z = suc z) → ∃x ∈ 𝜔 suc z = suc x)
23	19, 22	mpan2 403	. . . 4 ⊢ (z ∈ 𝜔 → ∃x ∈ 𝜔 suc z = suc x)
24	23	olcd 640	. . 3 ⊢ (z ∈ 𝜔 → (suc z = ∅ ∨ ∃x ∈ 𝜔 suc z = suc x))
25	24	a1d 22	. 2 ⊢ (z ∈ 𝜔 → ((z = ∅ ∨ ∃x ∈ 𝜔 z = suc x) → (suc z = ∅ ∨ ∃x ∈ 𝜔 suc z = suc x)))
26	4, 8, 12, 16, 18, 25	finds 4219	1 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∃x ∈ 𝜔 A = suc x))

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1373   ∈ wcel 1375  ∃wrex 2283  ∅c0 3202  suc csuc 4026  𝜔com 4209

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-nul 3835  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-iinf 4207

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-int 3568  df-suc 4031  df-iom 4210

This theorem is referenced by:  nnsuc  4234