Theorem onntri35 7155

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 7156), (3) implies (5) (onntri35 7155), (5) implies (1) (onntri51 7158), (2) implies (4) (onntri24 7160), (4) implies (5) (onntri45 7159), and (5) implies (2) (onntri52 7162).

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 7157 and exmidontri2or 7161, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7163 and (4) by onntri2or 7164.

(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 7144	. . . . 5 |- ~P 1o e. On
2	1	onsuci 4473	. . . 4 |- suc ~P 1o e. On
3		3on 6368	. . . 4 |- 3o e. On
4		eleq1 2220	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x e. y <-> suc ~P 1o e. y ) )
5		eqeq1 2164	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x = y <-> suc ~P 1o = y ) )
6		eleq2 2221	. . . . . . . 8 |- ( x = suc ~P 1o -> ( y e. x <-> y e. suc ~P 1o ) )
7	4, 5, 6	3orbi123d 1293	. . . . . . 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 657	. . . . . 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 657	. . . . 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 2221	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o e. y <-> suc ~P 1o e. 3o ) )
11		eqeq2 2167	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o = y <-> suc ~P 1o = 3o ) )
12		eleq1 2220	. . . . . . . 8 |- ( y = 3o -> ( y e. suc ~P 1o <-> 3o e. suc ~P 1o ) )
13	10, 11, 12	3orbi123d 1293	. . . . . . 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 657	. . . . . 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 657	. . . . 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 2829	. . . 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 423	. . 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 978	. . 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 666	. 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 7151	. . . 4 |- -. suc ~P 1o e. 3o
21	20	a1i 9	. . 3 |- ( -. EXMID -> -. suc ~P 1o e. 3o )
22		2on 6366	. . . . . . 7 |- 2o e. On
23		suc11 4515	. . . . . . 7 |- ( ( ~P 1o e. On /\ 2o e. On ) -> ( suc ~P 1o = suc 2o <-> ~P 1o = 2o ) )
24	1, 22, 23	mp2an 423	. . . . . 6 |- ( suc ~P 1o = suc 2o <-> ~P 1o = 2o )
25		df-3o 6359	. . . . . . 7 |- 3o = suc 2o
26	25	eqeq2i 2168	. . . . . 6 |- ( suc ~P 1o = 3o <-> suc ~P 1o = suc 2o )
27		exmidpweq 6847	. . . . . 6 |- (EXMID <-> ~P 1o = 2o )
28	24, 26, 27	3bitr4ri 212	. . . . 5 |- (EXMID <-> suc ~P 1o = 3o )
29	28	notbii 658	. . . 4 |- ( -. EXMID <-> -. suc ~P 1o = 3o )
30	29	biimpi 119	. . 3 |- ( -. EXMID -> -. suc ~P 1o = 3o )
31		3nelsucpw1 7152	. . . 4 |- -. 3o e. suc ~P 1o
32	31	a1i 9	. . 3 |- ( -. EXMID -> -. 3o e. suc ~P 1o )
33	21, 30, 32	3jca 1162	. 2 |- ( -. EXMID -> ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
34	19, 33	nsyl 618	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 962   /\ w3a 963   = wceq 1335   e. wcel 2128   A.wral 2435   ~Pcpw 3543  EXMIDwem 4154   Oncon0 4322   suc csuc 4324   1oc1o 6350   2oc2o 6351   3oc3o 6352

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-exmid 4155  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-1o 6357  df-2o 6358  df-3o 6359

This theorem is referenced by:  onntri3or  7163