Theorem onntri35 7383

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 7384), (3) implies (5) (onntri35 7383), (5) implies (1) (onntri51 7386), (2) implies (4) (onntri24 7388), (4) implies (5) (onntri45 7387), and (5) implies (2) (onntri52 7390).

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 7385 and exmidontri2or 7389, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7391 and (4) by onntri2or 7392.

(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 7372	. . . . 5 |- ~P 1o e. On
2	1	onsuci 4582	. . . 4 |- suc ~P 1o e. On
3		3on 6536	. . . 4 |- 3o e. On
4		eleq1 2270	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x e. y <-> suc ~P 1o e. y ) )
5		eqeq1 2214	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x = y <-> suc ~P 1o = y ) )
6		eleq2 2271	. . . . . . . 8 |- ( x = suc ~P 1o -> ( y e. x <-> y e. suc ~P 1o ) )
7	4, 5, 6	3orbi123d 1324	. . . . . . 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 669	. . . . . 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 669	. . . . 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 2271	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o e. y <-> suc ~P 1o e. 3o ) )
11		eqeq2 2217	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o = y <-> suc ~P 1o = 3o ) )
12		eleq1 2270	. . . . . . . 8 |- ( y = 3o -> ( y e. suc ~P 1o <-> 3o e. suc ~P 1o ) )
13	10, 11, 12	3orbi123d 1324	. . . . . . 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 669	. . . . . 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 669	. . . . 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 2897	. . . 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 996	. . 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 678	. 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 7379	. . . 4 |- -. suc ~P 1o e. 3o
21	20	a1i 9	. . 3 |- ( -. EXMID -> -. suc ~P 1o e. 3o )
22		2on 6534	. . . . . . 7 |- 2o e. On
23		suc11 4624	. . . . . . 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 6527	. . . . . . 7 |- 3o = suc 2o
26	25	eqeq2i 2218	. . . . . 6 |- ( suc ~P 1o = 3o <-> suc ~P 1o = suc 2o )
27		exmidpweq 7032	. . . . . 6 |- (EXMID <-> ~P 1o = 2o )
28	24, 26, 27	3bitr4ri 213	. . . . 5 |- (EXMID <-> suc ~P 1o = 3o )
29	28	notbii 670	. . . 4 |- ( -. EXMID <-> -. suc ~P 1o = 3o )
30	29	biimpi 120	. . 3 |- ( -. EXMID -> -. suc ~P 1o = 3o )
31		3nelsucpw1 7380	. . . 4 |- -. 3o e. suc ~P 1o
32	31	a1i 9	. . 3 |- ( -. EXMID -> -. 3o e. suc ~P 1o )
33	21, 30, 32	3jca 1180	. 2 |- ( -. EXMID -> ( -. suc ~P 1o e. 3o /\ -. suc ~P 1o = 3o /\ -. 3o e. suc ~P 1o ) )
34	19, 33	nsyl 629	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 980   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   ~Pcpw 3626  EXMIDwem 4254   Oncon0 4428   suc csuc 4430   1oc1o 6518   2oc2o 6519   3oc3o 6520

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-tr 4159  df-exmid 4255  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-1o 6525  df-2o 6526  df-3o 6527

This theorem is referenced by:  onntri3or  7391