Theorem nntri3or 6548

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 2257	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2203	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
3		eleq1 2256	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1322	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5	4	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)) ↔ (𝐴 ∈ ω → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))))
6		eleq2 2257	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
7		eqeq2 2203	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 = 𝑥 ↔ 𝐴 = ∅))
8		eleq1 2256	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	6, 7, 8	3orbi123d 1322	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10		eleq2 2257	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
11		eqeq2 2203	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = 𝑦))
12		eleq1 2256	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13	10, 11, 12	3orbi123d 1322	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
14		eleq2 2257	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
15		eqeq2 2203	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = suc 𝑦))
16		eleq1 2256	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
17	14, 15, 16	3orbi123d 1322	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
18		0elnn 4652	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19		olc 712	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20		3orass 983	. . . . . 6 ⊢ ((𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
21	19, 20	sylibr 134	. . . . 5 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
22	18, 21	syl 14	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴))
23		df-3or 981	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴))
24		elex 2771	. . . . . . . 8 ⊢ (𝑦 ∈ ω → 𝑦 ∈ V)
25		elsuc2g 4437	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦)))
26		3mix1 1168	. . . . . . . . 9 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
27	25, 26	biimtrrdi 164	. . . . . . . 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 6546	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30		elsuci 4435	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ suc 𝐴 → (suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴))
31	29, 30	biimtrdi 163	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 → (suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴)))
32		eqcom 2195	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 = 𝐴 ↔ 𝐴 = suc 𝑦)
33	32	orbi2i 763	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) ↔ (suc 𝑦 ∈ 𝐴 ∨ 𝐴 = suc 𝑦))
34	33	biimpi 120	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (suc 𝑦 ∈ 𝐴 ∨ 𝐴 = suc 𝑦))
35	34	orcomd 730	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
36	35	olcd 735	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
37		3orass 983	. . . . . . . . 9 ⊢ ((𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
38	36, 37	sylibr 134	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
39	31, 38	syl6 33	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
40	28, 39	jaao 720	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
41	23, 40	biimtrid 152	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) → (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
42	41	ex 115	. . . 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 4634	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)))
44	5, 43	vtoclga 2827	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
45	44	impcom 125	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   ∨ w3o 979   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ∅c0 3447  suc csuc 4397  ωcom 4623

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624

This theorem is referenced by:  nntri2  6549  nntri1  6551  nntri3  6552  nntri2or2  6553  nndceq  6554  nndcel  6555  nnsseleq  6556  nntr2  6558  nnawordex  6584  nnwetri  6974  nnnninfeq  7189  ltsopi  7382  pitri3or  7384  frec2uzlt2d  10478  nninfctlemfo  12180  ennnfonelemk  12560  ennnfonelemex  12574