Theorem nntri3or 6578

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 2268	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2214	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
3		eleq1 2267	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1323	. . . 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 2268	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
7		eqeq2 2214	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 = 𝑥 ↔ 𝐴 = ∅))
8		eleq1 2267	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	6, 7, 8	3orbi123d 1323	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10		eleq2 2268	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
11		eqeq2 2214	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = 𝑦))
12		eleq1 2267	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13	10, 11, 12	3orbi123d 1323	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
14		eleq2 2268	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
15		eqeq2 2214	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = suc 𝑦))
16		eleq1 2267	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
17	14, 15, 16	3orbi123d 1323	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
18		0elnn 4666	. . . . 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 2782	. . . . . . . 8 ⊢ (𝑦 ∈ ω → 𝑦 ∈ V)
25		elsuc2g 4451	. . . . . . . . 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 6576	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30		elsuci 4449	. . . . . . . . 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 2206	. . . . . . . . . . . . 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 4648	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)))
44	5, 43	vtoclga 2838	. 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 1372   ∈ wcel 2175  Vcvv 2771  ∅c0 3459  suc csuc 4411  ωcom 4637

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-tr 4142  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638

This theorem is referenced by:  nntri2  6579  nntri1  6581  nntri3  6582  nntri2or2  6583  nndceq  6584  nndcel  6585  nnsseleq  6586  nntr2  6588  nnawordex  6614  nnwetri  7012  nnnninfeq  7229  ltsopi  7432  pitri3or  7434  frec2uzlt2d  10547  nninfctlemfo  12303  ennnfonelemk  12713  ennnfonelemex  12727