Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sltsolem1 Unicode version

Theorem sltsolem1 24392
Description: Lemma for sltso 24393. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
Assertion
Ref Expression
sltsolem1  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } )

Proof of Theorem sltsolem1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 6510 . . . . . . . 8  |-  1o  =/=  (/)
2 df-ne 2461 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
31, 2mpbi 199 . . . . . . 7  |-  -.  1o  =  (/)
4 eqtr2 2314 . . . . . . 7  |-  ( ( x  =  1o  /\  x  =  (/) )  ->  1o  =  (/) )
53, 4mto 167 . . . . . 6  |-  -.  (
x  =  1o  /\  x  =  (/) )
6 1on 6502 . . . . . . . . 9  |-  1o  e.  On
7 0elon 4461 . . . . . . . . 9  |-  (/)  e.  On
8 df-2o 6496 . . . . . . . . . . 11  |-  2o  =  suc  1o
9 df-1o 6495 . . . . . . . . . . 11  |-  1o  =  suc  (/)
108, 9eqeq12i 2309 . . . . . . . . . 10  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
11 suc11 4512 . . . . . . . . . 10  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
1210, 11syl5bb 248 . . . . . . . . 9  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( 2o  =  1o  <->  1o  =  (/) ) )
136, 7, 12mp2an 653 . . . . . . . 8  |-  ( 2o  =  1o  <->  1o  =  (/) )
141, 13nemtbir 2547 . . . . . . 7  |-  -.  2o  =  1o
15 eqtr2 2314 . . . . . . . 8  |-  ( ( x  =  2o  /\  x  =  1o )  ->  2o  =  1o )
1615ancoms 439 . . . . . . 7  |-  ( ( x  =  1o  /\  x  =  2o )  ->  2o  =  1o )
1714, 16mto 167 . . . . . 6  |-  -.  (
x  =  1o  /\  x  =  2o )
18 nsuceq0 4488 . . . . . . . 8  |-  suc  1o  =/=  (/)
198eqeq1i 2303 . . . . . . . 8  |-  ( 2o  =  (/)  <->  suc  1o  =  (/) )
2018, 19nemtbir 2547 . . . . . . 7  |-  -.  2o  =  (/)
21 eqtr2 2314 . . . . . . . 8  |-  ( ( x  =  2o  /\  x  =  (/) )  ->  2o  =  (/) )
2221ancoms 439 . . . . . . 7  |-  ( ( x  =  (/)  /\  x  =  2o )  ->  2o  =  (/) )
2320, 22mto 167 . . . . . 6  |-  -.  (
x  =  (/)  /\  x  =  2o )
245, 17, 233pm3.2ni 24079 . . . . 5  |-  -.  (
( x  =  1o 
/\  x  =  (/) )  \/  ( x  =  1o  /\  x  =  2o )  \/  (
x  =  (/)  /\  x  =  2o ) )
25 vex 2804 . . . . . 6  |-  x  e. 
_V
2625, 25brtp 24176 . . . . 5  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x  <->  ( ( x  =  1o  /\  x  =  (/) )  \/  (
x  =  1o  /\  x  =  2o )  \/  ( x  =  (/)  /\  x  =  2o ) ) )
2724, 26mtbir 290 . . . 4  |-  -.  x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x
2827a1i 10 . . 3  |-  ( x  e.  { 1o ,  2o ,  (/) }  ->  -.  x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x )
29 vex 2804 . . . . . . 7  |-  y  e. 
_V
3025, 29brtp 24176 . . . . . 6  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  <->  ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
31 vex 2804 . . . . . . 7  |-  z  e. 
_V
3229, 31brtp 24176 . . . . . 6  |-  ( y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z  <->  ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) ) )
33 eqtr2 2314 . . . . . . . . . . . . 13  |-  ( ( y  =  1o  /\  y  =  (/) )  ->  1o  =  (/) )
343, 33mto 167 . . . . . . . . . . . 12  |-  -.  (
y  =  1o  /\  y  =  (/) )
3534pm2.21i 123 . . . . . . . . . . 11  |-  ( ( y  =  1o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
3635ad2ant2rl 729 . . . . . . . . . 10  |-  ( ( ( y  =  1o 
/\  z  =  (/) )  /\  ( x  =  1o  /\  y  =  (/) ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
3736expcom 424 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  1o  /\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
3835ad2ant2rl 729 . . . . . . . . . 10  |-  ( ( ( y  =  1o 
/\  z  =  2o )  /\  ( x  =  1o  /\  y  =  (/) ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
3938expcom 424 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  1o  /\  z  =  2o )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) ) )
40 3mix2 1125 . . . . . . . . . . 11  |-  ( ( x  =  1o  /\  z  =  2o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4140ad2ant2rl 729 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  (/) )  /\  ( y  =  (/)  /\  z  =  2o ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
4241ex 423 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  (/)  /\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
4337, 39, 423jaod 1246 . . . . . . . 8  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
44 eqtr2 2314 . . . . . . . . . . . . 13  |-  ( ( y  =  2o  /\  y  =  1o )  ->  2o  =  1o )
4514, 44mto 167 . . . . . . . . . . . 12  |-  -.  (
y  =  2o  /\  y  =  1o )
4645pm2.21i 123 . . . . . . . . . . 11  |-  ( ( y  =  2o  /\  y  =  1o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4746ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  1o  /\  z  =  (/) ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4847ex 423 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  1o  /\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
4946ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  1o  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5049ex 423 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  1o  /\  z  =  2o )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) ) )
51 eqtr2 2314 . . . . . . . . . . . . 13  |-  ( ( y  =  2o  /\  y  =  (/) )  ->  2o  =  (/) )
5220, 51mto 167 . . . . . . . . . . . 12  |-  -.  (
y  =  2o  /\  y  =  (/) )
5352pm2.21i 123 . . . . . . . . . . 11  |-  ( ( y  =  2o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5453ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  (/)  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5554ex 423 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  (/)  /\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
5648, 50, 553jaod 1246 . . . . . . . 8  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
5746ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  1o  /\  z  =  (/) ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
5857ex 423 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  1o 
/\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
5946ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  1o  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
6059ex 423 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  1o 
/\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6153ad2ant2lr 728 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  (/)  /\  z  =  2o ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
6261ex 423 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  (/)  /\  z  =  2o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6358, 60, 623jaod 1246 . . . . . . . 8  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6443, 56, 633jaoi 1245 . . . . . . 7  |-  ( ( ( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) )  -> 
( ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6564imp 418 . . . . . 6  |-  ( ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  /\  ( ( y  =  1o  /\  z  =  (/) )  \/  ( y  =  1o 
/\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) ) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
6630, 32, 65syl2anb 465 . . . . 5  |-  ( ( x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  /\  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
6725, 31brtp 24176 . . . . 5  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z  <->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
6866, 67sylibr 203 . . . 4  |-  ( ( x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  /\  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )  ->  x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )
6968a1i 10 . . 3  |-  ( ( x  e.  { 1o ,  2o ,  (/) }  /\  y  e.  { 1o ,  2o ,  (/) }  /\  z  e.  { 1o ,  2o ,  (/) } )  ->  ( ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  /\  y {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. } z )  ->  x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. } z ) )
7025eltp 3691 . . . . 5  |-  ( x  e.  { 1o ,  2o ,  (/) }  <->  ( x  =  1o  \/  x  =  2o  \/  x  =  (/) ) )
7129eltp 3691 . . . . 5  |-  ( y  e.  { 1o ,  2o ,  (/) }  <->  ( y  =  1o  \/  y  =  2o  \/  y  =  (/) ) )
72 eqtr3 2315 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  1o )  ->  x  =  y )
73723mix2d 24083 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  1o )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
7473ex 423 . . . . . . . 8  |-  ( x  =  1o  ->  (
y  =  1o  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
75 3mix2 1125 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
76753mix1d 24082 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
7776ex 423 . . . . . . . 8  |-  ( x  =  1o  ->  (
y  =  2o  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
78 3mix1 1124 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
79783mix1d 24082 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
8079ex 423 . . . . . . . 8  |-  ( x  =  1o  ->  (
y  =  (/)  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
8174, 77, 803jaod 1246 . . . . . . 7  |-  ( x  =  1o  ->  (
( y  =  1o  \/  y  =  2o  \/  y  =  (/) )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  ( x  =  1o 
/\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
82 3mix2 1125 . . . . . . . . . 10  |-  ( ( y  =  1o  /\  x  =  2o )  ->  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
83823mix3d 24084 . . . . . . . . 9  |-  ( ( y  =  1o  /\  x  =  2o )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
8483expcom 424 . . . . . . . 8  |-  ( x  =  2o  ->  (
y  =  1o  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
85 eqtr3 2315 . . . . . . . . . 10  |-  ( ( x  =  2o  /\  y  =  2o )  ->  x  =  y )
86853mix2d 24083 . . . . . . . . 9  |-  ( ( x  =  2o  /\  y  =  2o )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
8786ex 423 . . . . . . . 8  |-  ( x  =  2o  ->  (
y  =  2o  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
88 3mix3 1126 . . . . . . . . . 10  |-  ( ( y  =  (/)  /\  x  =  2o )  ->  (
( y  =  1o 
/\  x  =  (/) )  \/  ( y  =  1o  /\  x  =  2o )  \/  (
y  =  (/)  /\  x  =  2o ) ) )
89883mix3d 24084 . . . . . . . . 9  |-  ( ( y  =  (/)  /\  x  =  2o )  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
9089expcom 424 . . . . . . . 8  |-  ( x  =  2o  ->  (
y  =  (/)  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
9184, 87, 903jaod 1246 . . . . . . 7  |-  ( x  =  2o  ->  (
( y  =  1o  \/  y  =  2o  \/  y  =  (/) )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  ( x  =  1o 
/\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
92 3mix1 1124 . . . . . . . . . 10  |-  ( ( y  =  1o  /\  x  =  (/) )  -> 
( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
93923mix3d 24084 . . . . . . . . 9  |-  ( ( y  =  1o  /\  x  =  (/) )  -> 
( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
9493expcom 424 . . . . . . . 8  |-  ( x  =  (/)  ->  ( y  =  1o  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
95 3mix3 1126 . . . . . . . . . 10  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) ) )
96953mix1d 24082 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
9796ex 423 . . . . . . . 8  |-  ( x  =  (/)  ->  ( y  =  2o  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
98 eqtr3 2315 . . . . . . . . . 10  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  x  =  y )
99983mix2d 24083 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
10099ex 423 . . . . . . . 8  |-  ( x  =  (/)  ->  ( y  =  (/)  ->  ( ( ( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
10194, 97, 1003jaod 1246 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( y  =  1o  \/  y  =  2o  \/  y  =  (/) )  -> 
( ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
10281, 91, 1013jaoi 1245 . . . . . 6  |-  ( ( x  =  1o  \/  x  =  2o  \/  x  =  (/) )  -> 
( ( y  =  1o  \/  y  =  2o  \/  y  =  (/) )  ->  ( ( ( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) ) )
103102imp 418 . . . . 5  |-  ( ( ( x  =  1o  \/  x  =  2o  \/  x  =  (/) )  /\  ( y  =  1o  \/  y  =  2o  \/  y  =  (/) ) )  ->  (
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  ( y  =  1o 
/\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
10470, 71, 103syl2anb 465 . . . 4  |-  ( ( x  e.  { 1o ,  2o ,  (/) }  /\  y  e.  { 1o ,  2o ,  (/) } )  ->  ( ( ( x  =  1o  /\  y  =  (/) )  \/  ( x  =  1o 
/\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
105 biid 227 . . . . 5  |-  ( x  =  y  <->  x  =  y )
10629, 25brtp 24176 . . . . 5  |-  ( y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x  <->  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
10730, 105, 1063orbi123i 1141 . . . 4  |-  ( ( x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  \/  x  =  y  \/  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x )  <->  ( (
( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) )  \/  x  =  y  \/  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) ) )
108104, 107sylibr 203 . . 3  |-  ( ( x  e.  { 1o ,  2o ,  (/) }  /\  y  e.  { 1o ,  2o ,  (/) } )  ->  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. } y  \/  x  =  y  \/  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x ) )
10928, 69, 108issoi 4361 . 2  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  { 1o ,  2o ,  (/) }
110 df-tp 3661 . . 3  |-  { 1o ,  2o ,  (/) }  =  ( { 1o ,  2o }  u.  { (/) } )
111 soeq2 4350 . . 3  |-  ( { 1o ,  2o ,  (/)
}  =  ( { 1o ,  2o }  u.  { (/) } )  -> 
( { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  { 1o ,  2o ,  (/) }  <->  { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } ) ) )
112110, 111ax-mp 8 . 2  |-  ( {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  Or  { 1o ,  2o ,  (/)
}  <->  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
} ) )
113109, 112mpbi 199 1  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    u. cun 3163   (/)c0 3468   {csn 3653   {cpr 3654   {ctp 3655   <.cop 3656   class class class wbr 4039    Or wor 4329   Oncon0 4408   suc csuc 4410   1oc1o 6488   2oc2o 6489
This theorem is referenced by:  sltso  24393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-1o 6495  df-2o 6496
  Copyright terms: Public domain W3C validator