Theorem nntri3or 6637

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 2293	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2239	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
3		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1345	. . . 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 2293	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
7		eqeq2 2239	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 = 𝑥 ↔ 𝐴 = ∅))
8		eleq1 2292	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	6, 7, 8	3orbi123d 1345	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
10		eleq2 2293	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
11		eqeq2 2239	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = 𝑦))
12		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13	10, 11, 12	3orbi123d 1345	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
14		eleq2 2293	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
15		eqeq2 2239	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 = 𝑥 ↔ 𝐴 = suc 𝑦))
16		eleq1 2292	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
17	14, 15, 16	3orbi123d 1345	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ suc 𝑦 ∨ 𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
18		0elnn 4710	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
19		olc 716	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (𝐴 ∈ ∅ ∨ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
20		3orass 1005	. . . . . 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 1003	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) ∨ 𝑦 ∈ 𝐴))
24		elex 2811	. . . . . . . 8 ⊢ (𝑦 ∈ ω → 𝑦 ∈ V)
25		elsuc2g 4495	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (𝐴 ∈ suc 𝑦 ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦)))
26		3mix1 1190	. . . . . . . . 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 6635	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ suc 𝐴))
30		elsuci 4493	. . . . . . . . 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 2231	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 = 𝐴 ↔ 𝐴 = suc 𝑦)
33	32	orbi2i 767	. . . . . . . . . . . 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 734	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴))
36	35	olcd 739	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ 𝐴 ∨ suc 𝑦 = 𝐴) → (𝐴 ∈ suc 𝑦 ∨ (𝐴 = suc 𝑦 ∨ suc 𝑦 ∈ 𝐴)))
37		3orass 1005	. . . . . . . . 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 724	. . . . . 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 4692	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑥 ∨ 𝐴 = 𝑥 ∨ 𝑥 ∈ 𝐴)))
44	5, 43	vtoclga 2867	. 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 713   ∨ w3o 1001   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ∅c0 3491  suc csuc 4455  ωcom 4681

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 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3or 1003  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-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682

This theorem is referenced by:  nntri2  6638  nntri1  6640  nntri3  6641  nntri2or2  6642  nndceq  6643  nndcel  6644  nnsseleq  6645  nntr2  6647  nnawordex  6673  nnwetri  7074  nnnninfeq  7291  ltsopi  7503  pitri3or  7505  frec2uzlt2d  10621  nninfctlemfo  12556  ennnfonelemk  12966  ennnfonelemex  12980