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Theorem onntri35 7235
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 7236), (3) implies (5) (onntri35 7235), (5) implies (1) (onntri51 7238), (2) implies (4) (onntri24 7240), (4) implies (5) (onntri45 7239), and (5) implies (2) (onntri52 7242).

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 7237 and exmidontri2or 7241, respectively. Thus  -.  -. EXMID is equivalent to (1) or (2). In addition, 
-.  -. EXMID is equivalent to (3) by onntri3or 7243 and (4) by onntri2or 7244.

(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
StepHypRef Expression
1 pw1on 7224 . . . . 5  |-  ~P 1o  e.  On
21onsuci 4515 . . . 4  |-  suc  ~P 1o  e.  On
3 3on 6427 . . . 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 ) )
74, 5, 63orbi123d 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 ) ) )
87notbid 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 ) ) )
98notbid 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 ) )
1310, 11, 123orbi123d 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 ) ) )
1413notbid 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 ) ) )
1514notbid 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 ) ) )
169, 15rspc2v 2854 . . . 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 ) ) )
172, 3, 16mp2an 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 ) )
1917, 18sylnib 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 7231 . . . 4  |-  -.  suc  ~P 1o  e.  3o
2120a1i 9 . . 3  |-  ( -. EXMID  ->  -.  suc  ~P 1o  e.  3o )
22 2on 6425 . . . . . . 7  |-  2o  e.  On
23 suc11 4557 . . . . . . 7  |-  ( ( ~P 1o  e.  On  /\  2o  e.  On )  ->  ( suc  ~P 1o  =  suc  2o  <->  ~P 1o  =  2o ) )
241, 22, 23mp2an 426 . . . . . 6  |-  ( suc 
~P 1o  =  suc  2o  <->  ~P 1o  =  2o )
25 df-3o 6418 . . . . . . 7  |-  3o  =  suc  2o
2625eqeq2i 2188 . . . . . 6  |-  ( suc 
~P 1o  =  3o  <->  suc 
~P 1o  =  suc  2o )
27 exmidpweq 6908 . . . . . 6  |-  (EXMID  <->  ~P 1o  =  2o )
2824, 26, 273bitr4ri 213 . . . . 5  |-  (EXMID  <->  suc  ~P 1o  =  3o )
2928notbii 668 . . . 4  |-  ( -. EXMID  <->  -.  suc  ~P 1o  =  3o )
3029biimpi 120 . . 3  |-  ( -. EXMID  ->  -.  suc  ~P 1o  =  3o )
31 3nelsucpw1 7232 . . . 4  |-  -.  3o  e.  suc  ~P 1o
3231a1i 9 . . 3  |-  ( -. EXMID  ->  -.  3o  e.  suc  ~P 1o )
3321, 30, 323jca 1177 . 2  |-  ( -. EXMID  -> 
( -.  suc  ~P 1o  e.  3o  /\  -.  suc  ~P 1o  =  3o 
/\  -.  3o  e.  suc  ~P 1o ) )
3419, 33nsyl 628 1  |-  ( A. x  e.  On  A. y  e.  On  -.  -.  (
x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  -.  -. EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   ~Pcpw 3575  EXMIDwem 4194   Oncon0 4363   suc csuc 4365   1oc1o 6409   2oc2o 6410   3oc3o 6411
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-1o 6416  df-2o 6417  df-3o 6418
This theorem is referenced by:  onntri3or  7243
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