Theorem onntri35 7236

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 7237), (3) implies (5) (onntri35 7236), (5) implies (1) (onntri51 7239), (2) implies (4) (onntri24 7241), (4) implies (5) (onntri45 7240), and (5) implies (2) (onntri52 7243).

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 7238 and exmidontri2or 7242, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7244 and (4) by onntri2or 7245.

(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 7225	. . . . 5 |- ~P 1o e. On
2	1	onsuci 4516	. . . 4 |- suc ~P 1o e. On
3		3on 6428	. . . 4 |- 3o e. On
4		eleq1 2240	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x e. y <-> suc ~P 1o e. y ) )
5		eqeq1 2184	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x = y <-> suc ~P 1o = y ) )
6		eleq2 2241	. . . . . . . 8 |- ( x = suc ~P 1o -> ( y e. x <-> y e. suc ~P 1o ) )
7	4, 5, 6	3orbi123d 1311	. . . . . . 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 667	. . . . . 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 667	. . . . 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 2241	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o e. y <-> suc ~P 1o e. 3o ) )
11		eqeq2 2187	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o = y <-> suc ~P 1o = 3o ) )
12		eleq1 2240	. . . . . . . 8 |- ( y = 3o -> ( y e. suc ~P 1o <-> 3o e. suc ~P 1o ) )
13	10, 11, 12	3orbi123d 1311	. . . . . . 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 667	. . . . . 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 667	. . . . 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 2855	. . . 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 426	. . 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 993	. . 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 676	. 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 7232	. . . 4 |- -. suc ~P 1o e. 3o
21	20	a1i 9	. . 3 |- ( -. EXMID -> -. suc ~P 1o e. 3o )
22		2on 6426	. . . . . . 7 |- 2o e. On
23		suc11 4558	. . . . . . 7 |- ( ( ~P 1o e. On /\ 2o e. On ) -> ( suc ~P 1o = suc 2o <-> ~P 1o = 2o ) )
24	1, 22, 23	mp2an 426	. . . . . 6 |- ( suc ~P 1o = suc 2o <-> ~P 1o = 2o )
25		df-3o 6419	. . . . . . 7 |- 3o = suc 2o
26	25	eqeq2i 2188	. . . . . 6 |- ( suc ~P 1o = 3o <-> suc ~P 1o = suc 2o )
27		exmidpweq 6909	. . . . . 6 |- (EXMID <-> ~P 1o = 2o )
28	24, 26, 27	3bitr4ri 213	. . . . 5 |- (EXMID <-> suc ~P 1o = 3o )
29	28	notbii 668	. . . 4 |- ( -. EXMID <-> -. suc ~P 1o = 3o )
30	29	biimpi 120	. . 3 |- ( -. EXMID -> -. suc ~P 1o = 3o )
31		3nelsucpw1 7233	. . . 4 |- -. 3o e. suc ~P 1o
32	31	a1i 9	. . 3 |- ( -. EXMID -> -. 3o e. suc ~P 1o )
33	21, 30, 32	3jca 1177	. 2 |- ( -. EXMID -> ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
34	19, 33	nsyl 628	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 105   \/ w3o 977   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   ~Pcpw 3576  EXMIDwem 4195   Oncon0 4364   suc csuc 4366   1oc1o 6410   2oc2o 6411   3oc3o 6412

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-exmid 4196  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-1o 6417  df-2o 6418  df-3o 6419

This theorem is referenced by:  onntri3or  7244