Theorem nn0suc 4746

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 2245	. . 3 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
2		eqeq1 2245	. . . 4 ⊢ (𝑦 = ∅ → (𝑦 = suc 𝑥 ↔ ∅ = suc 𝑥))
3	2	rexbidv 2551	. . 3 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω ∅ = suc 𝑥))
4	1, 3	orbi12d 805	. 2 ⊢ (𝑦 = ∅ → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)))
5		eqeq1 2245	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
6		eqeq1 2245	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = suc 𝑥 ↔ 𝑧 = suc 𝑥))
7	6	rexbidv 2551	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑧 = suc 𝑥))
8	5, 7	orbi12d 805	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝑧 = ∅ ∨ ∃𝑥 ∈ ω 𝑧 = suc 𝑥)))
9		eqeq1 2245	. . 3 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
10		eqeq1 2245	. . . 4 ⊢ (𝑦 = suc 𝑧 → (𝑦 = suc 𝑥 ↔ suc 𝑧 = suc 𝑥))
11	10	rexbidv 2551	. . 3 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥))
12	9, 11	orbi12d 805	. 2 ⊢ (𝑦 = suc 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (suc 𝑧 = ∅ ∨ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)))
13		eqeq1 2245	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
14		eqeq1 2245	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
15	14	rexbidv 2551	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
16	13, 15	orbi12d 805	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
17		eqid 2238	. . 3 ⊢ ∅ = ∅
18	17	orci 743	. 2 ⊢ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)
19		eqid 2238	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
20		suceq 4542	. . . . . . 7 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21	20	eqeq2d 2250	. . . . . 6 ⊢ (𝑥 = 𝑧 → (suc 𝑧 = suc 𝑥 ↔ suc 𝑧 = suc 𝑧))
22	21	rspcev 2929	. . . . 5 ⊢ ((𝑧 ∈ ω ∧ suc 𝑧 = suc 𝑧) → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
23	19, 22	mpan2 429	. . . 4 ⊢ (𝑧 ∈ ω → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
24	23	olcd 746	. . 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 4742	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ∅c0 3520  suc csuc 4505  ωcom 4732

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 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-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733

This theorem is referenced by:  nnsuc  4758  nnpredcl  4765  frecabcl  6660  nnsucuniel  6758  nneneq  7148  phpm  7157  dif1enen  7174  fin0  7179  fin0or  7180  diffisn  7187