Theorem nntri3or 6072

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

Assertion

Ref	Expression
nntri3or	|- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. 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 e. x <-> A e. B ) )
2		eqeq2 2049	. . . . 5 |- ( x = B -> ( A = x <-> A = B ) )
3		eleq1 2100	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
4	1, 2, 3	3orbi123d 1206	. . . 4 |- ( x = B -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. B \/ A = B \/ B e. A ) ) )
5	4	imbi2d 219	. . 3 |- ( x = B -> ( ( A e. om -> ( A e. x \/ A = x \/ x e. A ) ) <-> ( A e. om -> ( A e. B \/ A = B \/ B e. A ) ) ) )
6		eleq2 2101	. . . . 5 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
7		eqeq2 2049	. . . . 5 |- ( x = (/) -> ( A = x <-> A = (/) ) )
8		eleq1 2100	. . . . 5 |- ( x = (/) -> ( x e. A <-> (/) e. A ) )
9	6, 7, 8	3orbi123d 1206	. . . 4 |- ( x = (/) -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. (/) \/ A = (/) \/ (/) e. A ) ) )
10		eleq2 2101	. . . . 5 |- ( x = y -> ( A e. x <-> A e. y ) )
11		eqeq2 2049	. . . . 5 |- ( x = y -> ( A = x <-> A = y ) )
12		eleq1 2100	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
13	10, 11, 12	3orbi123d 1206	. . . 4 |- ( x = y -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. y \/ A = y \/ y e. A ) ) )
14		eleq2 2101	. . . . 5 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
15		eqeq2 2049	. . . . 5 |- ( x = suc y -> ( A = x <-> A = suc y ) )
16		eleq1 2100	. . . . 5 |- ( x = suc y -> ( x e. A <-> suc y e. A ) )
17	14, 15, 16	3orbi123d 1206	. . . 4 |- ( x = suc y -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
18		0elnn 4340	. . . . 5 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
19		olc 632	. . . . . 6 |- ( ( A = (/) \/ (/) e. A ) -> ( A e. (/) \/ ( A = (/) \/ (/) e. A ) ) )
20		3orass 888	. . . . . 6 |- ( ( A e. (/) \/ A = (/) \/ (/) e. A ) <-> ( A e. (/) \/ ( A = (/) \/ (/) e. A ) ) )
21	19, 20	sylibr 137	. . . . 5 |- ( ( A = (/) \/ (/) e. A ) -> ( A e. (/) \/ A = (/) \/ (/) e. A ) )
22	18, 21	syl 14	. . . 4 |- ( A e. om -> ( A e. (/) \/ A = (/) \/ (/) e. A ) )
23		df-3or 886	. . . . . 6 |- ( ( A e. y \/ A = y \/ y e. A ) <-> ( ( A e. y \/ A = y ) \/ y e. A ) )
24		elex 2566	. . . . . . . 8 |- ( y e. om -> y e. _V )
25		elsuc2g 4142	. . . . . . . . 9 |- ( y e. _V -> ( A e. suc y <-> ( A e. y \/ A = y ) ) )
26		3mix1 1073	. . . . . . . . 9 |- ( A e. suc y -> ( A e. suc y \/ A = suc y \/ suc y e. A ) )
27	25, 26	syl6bir 153	. . . . . . . 8 |- ( y e. _V -> ( ( A e. y \/ A = y ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
28	24, 27	syl 14	. . . . . . 7 |- ( y e. om -> ( ( A e. y \/ A = y ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
29		nnsucelsuc 6070	. . . . . . . . 9 |- ( A e. om -> ( y e. A <-> suc y e. suc A ) )
30		elsuci 4140	. . . . . . . . 9 |- ( suc y e. suc A -> ( suc y e. A \/ suc y = A ) )
31	29, 30	syl6bi 152	. . . . . . . 8 |- ( A e. om -> ( y e. A -> ( suc y e. A \/ suc y = A ) ) )
32		eqcom 2042	. . . . . . . . . . . . 13 |- ( suc y = A <-> A = suc y )
33	32	orbi2i 679	. . . . . . . . . . . 12 |- ( ( suc y e. A \/ suc y = A ) <-> ( suc y e. A \/ A = suc y ) )
34	33	biimpi 113	. . . . . . . . . . 11 |- ( ( suc y e. A \/ suc y = A ) -> ( suc y e. A \/ A = suc y ) )
35	34	orcomd 648	. . . . . . . . . 10 |- ( ( suc y e. A \/ suc y = A ) -> ( A = suc y \/ suc y e. A ) )
36	35	olcd 653	. . . . . . . . 9 |- ( ( suc y e. A \/ suc y = A ) -> ( A e. suc y \/ ( A = suc y \/ suc y e. A ) ) )
37		3orass 888	. . . . . . . . 9 |- ( ( A e. suc y \/ A = suc y \/ suc y e. A ) <-> ( A e. suc y \/ ( A = suc y \/ suc y e. A ) ) )
38	36, 37	sylibr 137	. . . . . . . 8 |- ( ( suc y e. A \/ suc y = A ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) )
39	31, 38	syl6 29	. . . . . . 7 |- ( A e. om -> ( y e. A -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
40	28, 39	jaao 639	. . . . . 6 |- ( ( y e. om /\ A e. om ) -> ( ( ( A e. y \/ A = y ) \/ y e. A ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
41	23, 40	syl5bi 141	. . . . 5 |- ( ( y e. om /\ A e. om ) -> ( ( A e. y \/ A = y \/ y e. A ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
42	41	ex 108	. . . 4 |- ( y e. om -> ( A e. om -> ( ( A e. y \/ A = y \/ y e. A ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) ) )
43	9, 13, 17, 22, 42	finds2 4324	. . 3 |- ( x e. om -> ( A e. om -> ( A e. x \/ A = x \/ x e. A ) ) )
44	5, 43	vtoclga 2619	. 2 |- ( B e. om -> ( A e. om -> ( A e. B \/ A = B \/ B e. A ) ) )
45	44	impcom 116	1 |- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   \/ w3o 884   = wceq 1243   e. wcel 1393   _Vcvv 2557   (/)c0 3224   suc csuc 4102   omcom 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  6418  pitri3or  6420  frec2uzlt2d  9190