Theorem nntri3or 6382

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 2201	. . . . 5 |- ( x = B -> ( A e. x <-> A e. B ) )
2		eqeq2 2147	. . . . 5 |- ( x = B -> ( A = x <-> A = B ) )
3		eleq1 2200	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
4	1, 2, 3	3orbi123d 1289	. . . 4 |- ( x = B -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. B \/ A = B \/ B e. A ) ) )
5	4	imbi2d 229	. . 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 2201	. . . . 5 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
7		eqeq2 2147	. . . . 5 |- ( x = (/) -> ( A = x <-> A = (/) ) )
8		eleq1 2200	. . . . 5 |- ( x = (/) -> ( x e. A <-> (/) e. A ) )
9	6, 7, 8	3orbi123d 1289	. . . 4 |- ( x = (/) -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. (/) \/ A = (/) \/ (/) e. A ) ) )
10		eleq2 2201	. . . . 5 |- ( x = y -> ( A e. x <-> A e. y ) )
11		eqeq2 2147	. . . . 5 |- ( x = y -> ( A = x <-> A = y ) )
12		eleq1 2200	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
13	10, 11, 12	3orbi123d 1289	. . . 4 |- ( x = y -> ( ( A e. x \/ A = x \/ x e. A ) <-> ( A e. y \/ A = y \/ y e. A ) ) )
14		eleq2 2201	. . . . 5 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
15		eqeq2 2147	. . . . 5 |- ( x = suc y -> ( A = x <-> A = suc y ) )
16		eleq1 2200	. . . . 5 |- ( x = suc y -> ( x e. A <-> suc y e. A ) )
17	14, 15, 16	3orbi123d 1289	. . . 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 4527	. . . . 5 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
19		olc 700	. . . . . 6 |- ( ( A = (/) \/ (/) e. A ) -> ( A e. (/) \/ ( A = (/) \/ (/) e. A ) ) )
20		3orass 965	. . . . . 6 |- ( ( A e. (/) \/ A = (/) \/ (/) e. A ) <-> ( A e. (/) \/ ( A = (/) \/ (/) e. A ) ) )
21	19, 20	sylibr 133	. . . . 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 963	. . . . . 6 |- ( ( A e. y \/ A = y \/ y e. A ) <-> ( ( A e. y \/ A = y ) \/ y e. A ) )
24		elex 2692	. . . . . . . 8 |- ( y e. om -> y e. _V )
25		elsuc2g 4322	. . . . . . . . 9 |- ( y e. _V -> ( A e. suc y <-> ( A e. y \/ A = y ) ) )
26		3mix1 1150	. . . . . . . . 9 |- ( A e. suc y -> ( A e. suc y \/ A = suc y \/ suc y e. A ) )
27	25, 26	syl6bir 163	. . . . . . . 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 6380	. . . . . . . . 9 |- ( A e. om -> ( y e. A <-> suc y e. suc A ) )
30		elsuci 4320	. . . . . . . . 9 |- ( suc y e. suc A -> ( suc y e. A \/ suc y = A ) )
31	29, 30	syl6bi 162	. . . . . . . 8 |- ( A e. om -> ( y e. A -> ( suc y e. A \/ suc y = A ) ) )
32		eqcom 2139	. . . . . . . . . . . . 13 |- ( suc y = A <-> A = suc y )
33	32	orbi2i 751	. . . . . . . . . . . 12 |- ( ( suc y e. A \/ suc y = A ) <-> ( suc y e. A \/ A = suc y ) )
34	33	biimpi 119	. . . . . . . . . . 11 |- ( ( suc y e. A \/ suc y = A ) -> ( suc y e. A \/ A = suc y ) )
35	34	orcomd 718	. . . . . . . . . 10 |- ( ( suc y e. A \/ suc y = A ) -> ( A = suc y \/ suc y e. A ) )
36	35	olcd 723	. . . . . . . . 9 |- ( ( suc y e. A \/ suc y = A ) -> ( A e. suc y \/ ( A = suc y \/ suc y e. A ) ) )
37		3orass 965	. . . . . . . . 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 133	. . . . . . . 8 |- ( ( suc y e. A \/ suc y = A ) -> ( A e. suc y \/ A = suc y \/ suc y e. A ) )
39	31, 38	syl6 33	. . . . . . 7 |- ( A e. om -> ( y e. A -> ( A e. suc y \/ A = suc y \/ suc y e. A ) ) )
40	28, 39	jaao 708	. . . . . 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 151	. . . . 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 114	. . . 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 4510	. . 3 |- ( x e. om -> ( A e. om -> ( A e. x \/ A = x \/ x e. A ) ) )
44	5, 43	vtoclga 2747	. 2 |- ( B e. om -> ( A e. om -> ( A e. B \/ A = B \/ B e. A ) ) )
45	44	impcom 124	1 |- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   \/ w3o 961   = wceq 1331   e. wcel 1480   _Vcvv 2681   (/)c0 3358   suc csuc 4282   omcom 4499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500

This theorem is referenced by:  nntri2  6383  nntri1  6385  nntri3  6386  nntri2or2  6387  nndceq  6388  nndcel  6389  nnsseleq  6390  nntr2  6392  nnawordex  6417  nnwetri  6797  ltsopi  7121  pitri3or  7123  frec2uzlt2d  10170  ennnfonelemk  11902  ennnfonelemex  11916  nninfalllemn  13191