Theorem nn0suc 4693

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 2236	. . 3 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
2		eqeq1 2236	. . . 4 ⊢ (𝑦 = ∅ → (𝑦 = suc 𝑥 ↔ ∅ = suc 𝑥))
3	2	rexbidv 2531	. . 3 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω ∅ = suc 𝑥))
4	1, 3	orbi12d 798	. 2 ⊢ (𝑦 = ∅ → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)))
5		eqeq1 2236	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
6		eqeq1 2236	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = suc 𝑥 ↔ 𝑧 = suc 𝑥))
7	6	rexbidv 2531	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑧 = suc 𝑥))
8	5, 7	orbi12d 798	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝑧 = ∅ ∨ ∃𝑥 ∈ ω 𝑧 = suc 𝑥)))
9		eqeq1 2236	. . 3 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
10		eqeq1 2236	. . . 4 ⊢ (𝑦 = suc 𝑧 → (𝑦 = suc 𝑥 ↔ suc 𝑧 = suc 𝑥))
11	10	rexbidv 2531	. . 3 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥))
12	9, 11	orbi12d 798	. 2 ⊢ (𝑦 = suc 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (suc 𝑧 = ∅ ∨ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)))
13		eqeq1 2236	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
14		eqeq1 2236	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = suc 𝑥 ↔ 𝐴 = suc 𝑥))
15	14	rexbidv 2531	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
16	13, 15	orbi12d 798	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
17		eqid 2229	. . 3 ⊢ ∅ = ∅
18	17	orci 736	. 2 ⊢ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)
19		eqid 2229	. . . . 5 ⊢ suc 𝑧 = suc 𝑧
20		suceq 4490	. . . . . . 7 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21	20	eqeq2d 2241	. . . . . 6 ⊢ (𝑥 = 𝑧 → (suc 𝑧 = suc 𝑥 ↔ suc 𝑧 = suc 𝑧))
22	21	rspcev 2907	. . . . 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 739	. . 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 4689	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  ∅c0 3491  suc csuc 4453  ωcom 4679

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-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  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-uni 3888  df-int 3923  df-suc 4459  df-iom 4680

This theorem is referenced by:  nnsuc  4705  nnpredcl  4712  frecabcl  6535  nnsucuniel  6631  nneneq  7006  phpm  7015  dif1enen  7030  fin0  7035  fin0or  7036  diffisn  7043