Theorem onntri35 7214

Description: Double negated ordinal trichotomy.

There are five equivalent statements: (1) -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ), (2) -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ), (3) A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ), (4) A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ), and (5) -. -. EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7215), (3) implies (5) (onntri35 7214), (5) implies (1) (onntri51 7217), (2) implies (4) (onntri24 7219), (4) implies (5) (onntri45 7218), and (5) implies (2) (onntri52 7221).

Another way of stating this is that EXMID is equivalent to trichotomy, either the x e. y \/ x = y \/ y e. x or the x C_ y \/ y C_ x form, as shown in exmidontri 7216 and exmidontri2or 7220, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7222 and (4) by onntri2or 7223.

(Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

Assertion

Ref	Expression
onntri35	|- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. EXMID )

Distinct variable group:   x, y

Proof of Theorem onntri35

Step	Hyp	Ref	Expression
1		pw1on 7203	. . . . 5 |- ~P 1o e. On
2	1	onsuci 4500	. . . 4 |- suc ~P 1o e. On
3		3on 6406	. . . 4 |- 3o e. On
4		eleq1 2233	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x e. y <-> suc ~P 1o e. y ) )
5		eqeq1 2177	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x = y <-> suc ~P 1o = y ) )
6		eleq2 2234	. . . . . . . 8 |- ( x = suc ~P 1o -> ( y e. x <-> y e. suc ~P 1o ) )
7	4, 5, 6	3orbi123d 1306	. . . . . . 7 |- ( x = suc ~P 1o -> ( ( x e. y \/ x = y \/ y e. x ) <-> ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) ) )
8	7	notbid 662	. . . . . 6 |- ( x = suc ~P 1o -> ( -. ( x e. y \/ x = y \/ y e. x ) <-> -. ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) ) )
9	8	notbid 662	. . . . 5 |- ( x = suc ~P 1o -> ( -. -. ( x e. y \/ x = y \/ y e. x ) <-> -. -. ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) ) )
10		eleq2 2234	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o e. y <-> suc ~P 1o e. 3o ) )
11		eqeq2 2180	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o = y <-> suc ~P 1o = 3o ) )
12		eleq1 2233	. . . . . . . 8 |- ( y = 3o -> ( y e. suc ~P 1o <-> 3o e. suc ~P 1o ) )
13	10, 11, 12	3orbi123d 1306	. . . . . . 7 |- ( y = 3o -> ( ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) <-> ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) ) )
14	13	notbid 662	. . . . . 6 |- ( y = 3o -> ( -. ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) <-> -. ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) ) )
15	14	notbid 662	. . . . 5 |- ( y = 3o -> ( -. -. ( suc ~P 1o e. y \/ suc ~P 1o = y \/ y e. suc ~P 1o ) <-> -. -. ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) ) )
16	9, 15	rspc2v 2847	. . . 4 |- ( ( suc ~P 1o e. On /\ 3o e. On ) -> ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) ) )
17	2, 3, 16	mp2an 424	. . 3 |- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) )
18		3ioran 988	. . 3 |- ( -. ( suc ~P 1o e. 3o \/ suc ~P 1o = 3o \/ 3o e. suc ~P 1o ) <-> ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
19	17, 18	sylnib 671	. 2 |- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
20		sucpw1nel3 7210	. . . 4 |- -. suc ~P 1o e. 3o
21	20	a1i 9	. . 3 |- ( -. EXMID -> -. suc ~P 1o e. 3o )
22		2on 6404	. . . . . . 7 |- 2o e. On
23		suc11 4542	. . . . . . 7 |- ( ( ~P 1o e. On /\ 2o e. On ) -> ( suc ~P 1o = suc 2o <-> ~P 1o = 2o ) )
24	1, 22, 23	mp2an 424	. . . . . 6 |- ( suc ~P 1o = suc 2o <-> ~P 1o = 2o )
25		df-3o 6397	. . . . . . 7 |- 3o = suc 2o
26	25	eqeq2i 2181	. . . . . 6 |- ( suc ~P 1o = 3o <-> suc ~P 1o = suc 2o )
27		exmidpweq 6887	. . . . . 6 |- (EXMID <-> ~P 1o = 2o )
28	24, 26, 27	3bitr4ri 212	. . . . 5 |- (EXMID <-> suc ~P 1o = 3o )
29	28	notbii 663	. . . 4 |- ( -. EXMID <-> -. suc ~P 1o = 3o )
30	29	biimpi 119	. . 3 |- ( -. EXMID -> -. suc ~P 1o = 3o )
31		3nelsucpw1 7211	. . . 4 |- -. 3o e. suc ~P 1o
32	31	a1i 9	. . 3 |- ( -. EXMID -> -. 3o e. suc ~P 1o )
33	21, 30, 32	3jca 1172	. 2 |- ( -. EXMID -> ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
34	19, 33	nsyl 623	1 |- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. EXMID )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ w3o 972   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   ~Pcpw 3566  EXMIDwem 4180   Oncon0 4348   suc csuc 4350   1oc1o 6388   2oc2o 6389   3oc3o 6390

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-1o 6395  df-2o 6396  df-3o 6397

This theorem is referenced by:  onntri3or  7222