Theorem nn0suc 4641

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 2203	. . 3 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
2		eqeq1 2203	. . . 4 ⊢ (𝑦 = ∅ → (𝑦 = suc 𝑥 ↔ ∅ = suc 𝑥))
3	2	rexbidv 2498	. . 3 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω ∅ = suc 𝑥))
4	1, 3	orbi12d 794	. 2 ⊢ (𝑦 = ∅ → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)))
5		eqeq1 2203	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
6		eqeq1 2203	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = suc 𝑥 ↔ 𝑧 = suc 𝑥))
7	6	rexbidv 2498	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑧 = suc 𝑥))
8	5, 7	orbi12d 794	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝑧 = ∅ ∨ ∃𝑥 ∈ ω 𝑧 = suc 𝑥)))
9		eqeq1 2203	. . 3 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
10		eqeq1 2203	. . . 4 ⊢ (𝑦 = suc 𝑧 → (𝑦 = suc 𝑥 ↔ suc 𝑧 = suc 𝑥))
11	10	rexbidv 2498	. . 3 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥))
12	9, 11	orbi12d 794	. 2 ⊢ (𝑦 = suc 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (suc 𝑧 = ∅ ∨ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)))
13		eqeq1 2203	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
14		eqeq1 2203	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
15	14	rexbidv 2498	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
16	13, 15	orbi12d 794	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
17		eqid 2196	. . 3 ⊢ ∅ = ∅
18	17	orci 732	. 2 ⊢ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)
19		eqid 2196	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
20		suceq 4438	. . . . . . 7 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21	20	eqeq2d 2208	. . . . . 6 ⊢ (𝑥 = 𝑧 → (suc 𝑧 = suc 𝑥 ↔ suc 𝑧 = suc 𝑧))
22	21	rspcev 2868	. . . . 5 ⊢ ((𝑧 ∈ ω ∧ suc 𝑧 = suc 𝑧) → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
23	19, 22	mpan2 425	. . . 4 ⊢ (𝑧 ∈ ω → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
24	23	olcd 735	. . 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 4637	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ∨ wo 709   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  ∅c0 3451  suc csuc 4401  ωcom 4627

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628

This theorem is referenced by:  nnsuc  4653  nnpredcl  4660  frecabcl  6466  nnsucuniel  6562  nneneq  6927  phpm  6935  dif1enen  6950  fin0  6955  fin0or  6956  diffisn  6963