Theorem nntri3or 6592

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 2270	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2216	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
3		eleq1 2269	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1324	. . . 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 2270	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
7		eqeq2 2216	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 = 𝑥 ↔ 𝐴 = ∅))
8		eleq1 2269	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	6, 7, 8	3orbi123d 1324	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10		eleq2 2270	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
11		eqeq2 2216	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = 𝑦))
12		eleq1 2269	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13	10, 11, 12	3orbi123d 1324	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
14		eleq2 2270	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
15		eqeq2 2216	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = suc 𝑦))
16		eleq1 2269	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
17	14, 15, 16	3orbi123d 1324	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
18		0elnn 4675	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19		olc 713	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20		3orass 984	. . . . . 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 982	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴))
24		elex 2785	. . . . . . . 8 ⊢ (𝑦 ∈ ω → 𝑦 ∈ V)
25		elsuc2g 4460	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦)))
26		3mix1 1169	. . . . . . . . 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 6590	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30		elsuci 4458	. . . . . . . . 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 2208	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 = 𝐴 ↔ 𝐴 = suc 𝑦)
33	32	orbi2i 764	. . . . . . . . . . . 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 731	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
36	35	olcd 736	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
37		3orass 984	. . . . . . . . 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 721	. . . . . 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 4657	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)))
44	5, 43	vtoclga 2841	. 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 710   ∨ w3o 980   = wceq 1373   ∈ wcel 2177  Vcvv 2773  ∅c0 3464  suc csuc 4420  ωcom 4646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-tr 4151  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647

This theorem is referenced by:  nntri2  6593  nntri1  6595  nntri3  6596  nntri2or2  6597  nndceq  6598  nndcel  6599  nnsseleq  6600  nntr2  6602  nnawordex  6628  nnwetri  7028  nnnninfeq  7245  ltsopi  7453  pitri3or  7455  frec2uzlt2d  10571  nninfctlemfo  12436  ennnfonelemk  12846  ennnfonelemex  12860