Theorem onntri35 7351

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 7352), (3) implies (5) (onntri35 7351), (5) implies (1) (onntri51 7354), (2) implies (4) (onntri24 7356), (4) implies (5) (onntri45 7355), and (5) implies (2) (onntri52 7358).

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 7353 and exmidontri2or 7357, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7359 and (4) by onntri2or 7360.

(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 7340	. . . . 5 |- ~P 1o e. On
2	1	onsuci 4565	. . . 4 |- suc ~P 1o e. On
3		3on 6515	. . . 4 |- 3o e. On
4		eleq1 2268	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x e. y <-> suc ~P 1o e. y ) )
5		eqeq1 2212	. . . . . . . 8 |- ( x = suc ~P 1o -> ( x = y <-> suc ~P 1o = y ) )
6		eleq2 2269	. . . . . . . 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 2269	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o e. y <-> suc ~P 1o e. 3o ) )
11		eqeq2 2215	. . . . . . . 8 |- ( y = 3o -> ( suc ~P 1o = y <-> suc ~P 1o = 3o ) )
12		eleq1 2268	. . . . . . . 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 2890	. . . 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 7347	. . . 4 |- -. suc ~P 1o e. 3o
21	20	a1i 9	. . 3 |- ( -. EXMID -> -. suc ~P 1o e. 3o )
22		2on 6513	. . . . . . 7 |- 2o e. On
23		suc11 4607	. . . . . . 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 6506	. . . . . . 7 |- 3o = suc 2o
26	25	eqeq2i 2216	. . . . . 6 |- ( suc ~P 1o = 3o <-> suc ~P 1o = suc 2o )
27		exmidpweq 7008	. . . . . 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 7348	. . . 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 2176   A.wral 2484   ~Pcpw 3616  EXMIDwem 4239   Oncon0 4411   suc csuc 4413   1oc1o 6497   2oc2o 6498   3oc3o 6499

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4144  df-exmid 4240  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-1o 6504  df-2o 6505  df-3o 6506

This theorem is referenced by:  onntri3or  7359