Theorem nntri3or 6072

Description: Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)

Assertion

Ref	Expression
nntri3or	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Proof of Theorem nntri3or

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

Step	Hyp	Ref	Expression
1		eleq2 2101	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2049	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
3		eleq1 2100	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1206	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5	4	imbi2d 219	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)) ↔ (𝐴 ∈ ω → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))))
6		eleq2 2101	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
7		eqeq2 2049	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 = 𝑥 ↔ 𝐴 = ∅))
8		eleq1 2100	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	6, 7, 8	3orbi123d 1206	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10		eleq2 2101	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
11		eqeq2 2049	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = 𝑦))
12		eleq1 2100	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13	10, 11, 12	3orbi123d 1206	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
14		eleq2 2101	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
15		eqeq2 2049	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = suc 𝑦))
16		eleq1 2100	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
17	14, 15, 16	3orbi123d 1206	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
18		0elnn 4340	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19		olc 632	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20		3orass 888	. . . . . 6 ⊢ ((𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
21	19, 20	sylibr 137	. . . . 5 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
22	18, 21	syl 14	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
23		df-3or 886	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴))
24		elex 2566	. . . . . . . 8 ⊢ (𝑦 ∈ ω → 𝑦 ∈ V)
25		elsuc2g 4142	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦)))
26		3mix1 1073	. . . . . . . . 9 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
27	25, 26	syl6bir 153	. . . . . . . 8 ⊢ (𝑦 ∈ V → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
28	24, 27	syl 14	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
29		nnsucelsuc 6070	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30		elsuci 4140	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ suc 𝐴 → (suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴))
31	29, 30	syl6bi 152	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 → (suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴)))
32		eqcom 2042	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 = 𝐴 ↔ 𝐴 = suc 𝑦)
33	32	orbi2i 679	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) ↔ (suc 𝑦 ∈ 𝐴 ∨ 𝐴 = suc 𝑦))
34	33	biimpi 113	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (suc 𝑦 ∈ 𝐴 ∨ 𝐴 = suc 𝑦))
35	34	orcomd 648	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
36	35	olcd 653	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
37		3orass 888	. . . . . . . . 9 ⊢ ((𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
38	36, 37	sylibr 137	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
39	31, 38	syl6 29	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
40	28, 39	jaao 639	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
41	23, 40	syl5bi 141	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
42	41	ex 108	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))))
43	9, 13, 17, 22, 42	finds2 4324	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)))
44	5, 43	vtoclga 2619	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
45	44	impcom 116	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629   ∨ w3o 884   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  suc csuc 4102  ωcom 4313

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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314

This theorem is referenced by:  nntri2  6073  nntri1  6074  nntri3  6075  nntri2or2  6076  nndceq  6077  nndcel  6078  nnsseleq  6079  nnawordex  6101  nnwetri  6354  ltsopi  6416  pitri3or  6418  frec2uzlt2d  9164