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Theorem sltsolem1 29602
Description: Lemma for sltso 29603. 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 7163 . . . . . . . 8  |-  1o  =/=  (/)
21neii 2656 . . . . . . 7  |-  -.  1o  =  (/)
3 eqtr2 2484 . . . . . . 7  |-  ( ( x  =  1o  /\  x  =  (/) )  ->  1o  =  (/) )
42, 3mto 176 . . . . . 6  |-  -.  (
x  =  1o  /\  x  =  (/) )
5 1on 7155 . . . . . . . . 9  |-  1o  e.  On
6 0elon 4940 . . . . . . . . 9  |-  (/)  e.  On
7 df-2o 7149 . . . . . . . . . . 11  |-  2o  =  suc  1o
8 df-1o 7148 . . . . . . . . . . 11  |-  1o  =  suc  (/)
97, 8eqeq12i 2477 . . . . . . . . . 10  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
10 suc11 4990 . . . . . . . . . 10  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
119, 10syl5bb 257 . . . . . . . . 9  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( 2o  =  1o  <->  1o  =  (/) ) )
125, 6, 11mp2an 672 . . . . . . . 8  |-  ( 2o  =  1o  <->  1o  =  (/) )
131, 12nemtbir 2785 . . . . . . 7  |-  -.  2o  =  1o
14 eqtr2 2484 . . . . . . . 8  |-  ( ( x  =  2o  /\  x  =  1o )  ->  2o  =  1o )
1514ancoms 453 . . . . . . 7  |-  ( ( x  =  1o  /\  x  =  2o )  ->  2o  =  1o )
1613, 15mto 176 . . . . . 6  |-  -.  (
x  =  1o  /\  x  =  2o )
17 nsuceq0 4967 . . . . . . . 8  |-  suc  1o  =/=  (/)
187eqeq1i 2464 . . . . . . . 8  |-  ( 2o  =  (/)  <->  suc  1o  =  (/) )
1917, 18nemtbir 2785 . . . . . . 7  |-  -.  2o  =  (/)
20 eqtr2 2484 . . . . . . . 8  |-  ( ( x  =  2o  /\  x  =  (/) )  ->  2o  =  (/) )
2120ancoms 453 . . . . . . 7  |-  ( ( x  =  (/)  /\  x  =  2o )  ->  2o  =  (/) )
2219, 21mto 176 . . . . . 6  |-  -.  (
x  =  (/)  /\  x  =  2o )
234, 16, 223pm3.2ni 29265 . . . . 5  |-  -.  (
( x  =  1o 
/\  x  =  (/) )  \/  ( x  =  1o  /\  x  =  2o )  \/  (
x  =  (/)  /\  x  =  2o ) )
24 vex 3112 . . . . . 6  |-  x  e. 
_V
2524, 24brtp 29353 . . . . 5  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x  <->  ( ( x  =  1o  /\  x  =  (/) )  \/  (
x  =  1o  /\  x  =  2o )  \/  ( x  =  (/)  /\  x  =  2o ) ) )
2623, 25mtbir 299 . . . 4  |-  -.  x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x
2726a1i 11 . . 3  |-  ( x  e.  { 1o ,  2o ,  (/) }  ->  -.  x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x )
28 vex 3112 . . . . . . 7  |-  y  e. 
_V
2924, 28brtp 29353 . . . . . 6  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  <->  ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
30 vex 3112 . . . . . . 7  |-  z  e. 
_V
3128, 30brtp 29353 . . . . . 6  |-  ( y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z  <->  ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) ) )
32 eqtr2 2484 . . . . . . . . . . . . 13  |-  ( ( y  =  1o  /\  y  =  (/) )  ->  1o  =  (/) )
332, 32mto 176 . . . . . . . . . . . 12  |-  -.  (
y  =  1o  /\  y  =  (/) )
3433pm2.21i 131 . . . . . . . . . . 11  |-  ( ( y  =  1o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
3534ad2ant2rl 748 . . . . . . . . . 10  |-  ( ( ( y  =  1o 
/\  z  =  (/) )  /\  ( x  =  1o  /\  y  =  (/) ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
3635expcom 435 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  1o  /\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
3734ad2ant2rl 748 . . . . . . . . . 10  |-  ( ( ( y  =  1o 
/\  z  =  2o )  /\  ( x  =  1o  /\  y  =  (/) ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
3837expcom 435 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  1o  /\  z  =  2o )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) ) )
39 3mix2 1166 . . . . . . . . . . 11  |-  ( ( x  =  1o  /\  z  =  2o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4039ad2ant2rl 748 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  (/) )  /\  ( y  =  (/)  /\  z  =  2o ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
4140ex 434 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( y  =  (/)  /\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
4236, 38, 413jaod 1292 . . . . . . . 8  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
43 eqtr2 2484 . . . . . . . . . . . . 13  |-  ( ( y  =  2o  /\  y  =  1o )  ->  2o  =  1o )
4413, 43mto 176 . . . . . . . . . . . 12  |-  -.  (
y  =  2o  /\  y  =  1o )
4544pm2.21i 131 . . . . . . . . . . 11  |-  ( ( y  =  2o  /\  y  =  1o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4645ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  1o  /\  z  =  (/) ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4746ex 434 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  1o  /\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
4845ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  1o  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
4948ex 434 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  1o  /\  z  =  2o )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) ) )
50 eqtr2 2484 . . . . . . . . . . . . 13  |-  ( ( y  =  2o  /\  y  =  (/) )  ->  2o  =  (/) )
5119, 50mto 176 . . . . . . . . . . . 12  |-  -.  (
y  =  2o  /\  y  =  (/) )
5251pm2.21i 131 . . . . . . . . . . 11  |-  ( ( y  =  2o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5352ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  1o 
/\  y  =  2o )  /\  ( y  =  (/)  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5453ex 434 . . . . . . . . 9  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( y  =  (/)  /\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
5547, 49, 543jaod 1292 . . . . . . . 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 ) ) ) )
5645ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  1o  /\  z  =  (/) ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
5756ex 434 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  1o 
/\  z  =  (/) )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
5845ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  1o  /\  z  =  2o ) )  -> 
( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
5958ex 434 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  1o 
/\  z  =  2o )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6052ad2ant2lr 747 . . . . . . . . . 10  |-  ( ( ( x  =  (/)  /\  y  =  2o )  /\  ( y  =  (/)  /\  z  =  2o ) )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
6160ex 434 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( y  =  (/)  /\  z  =  2o )  ->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6257, 59, 613jaod 1292 . . . . . . . 8  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) )  ->  ( (
x  =  1o  /\  z  =  (/) )  \/  ( x  =  1o 
/\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) ) )
6342, 55, 623jaoi 1291 . . . . . . 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 ) ) ) )
6463imp 429 . . . . . 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 ) ) )
6529, 31, 64syl2anb 479 . . . . 5  |-  ( ( x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  /\  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )  ->  (
( x  =  1o 
/\  z  =  (/) )  \/  ( x  =  1o  /\  z  =  2o )  \/  (
x  =  (/)  /\  z  =  2o ) ) )
6624, 30brtp 29353 . . . . 5  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z  <->  ( ( x  =  1o  /\  z  =  (/) )  \/  (
x  =  1o  /\  z  =  2o )  \/  ( x  =  (/)  /\  z  =  2o ) ) )
6765, 66sylibr 212 . . . 4  |-  ( ( x { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  /\  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )  ->  x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z )
6867a1i 11 . . 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 ) )
6924eltp 4077 . . . . 5  |-  ( x  e.  { 1o ,  2o ,  (/) }  <->  ( x  =  1o  \/  x  =  2o  \/  x  =  (/) ) )
7028eltp 4077 . . . . 5  |-  ( y  e.  { 1o ,  2o ,  (/) }  <->  ( y  =  1o  \/  y  =  2o  \/  y  =  (/) ) )
71 eqtr3 2485 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  1o )  ->  x  =  y )
72713mix2d 1172 . . . . . . . . 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 ) ) ) )
7372ex 434 . . . . . . . 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 ) ) ) ) )
74 3mix2 1166 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  2o )  ->  ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
75743mix1d 1171 . . . . . . . . 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 ) ) ) )
7675ex 434 . . . . . . . 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 ) ) ) ) )
77 3mix1 1165 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  (/) )  -> 
( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
78773mix1d 1171 . . . . . . . . 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 ) ) ) )
7978ex 434 . . . . . . . 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 ) ) ) ) )
8073, 76, 793jaod 1292 . . . . . . 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 ) ) ) ) )
81 3mix2 1166 . . . . . . . . . 10  |-  ( ( y  =  1o  /\  x  =  2o )  ->  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
82813mix3d 1173 . . . . . . . . 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 ) ) ) )
8382expcom 435 . . . . . . . 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 ) ) ) ) )
84 eqtr3 2485 . . . . . . . . . 10  |-  ( ( x  =  2o  /\  y  =  2o )  ->  x  =  y )
85843mix2d 1172 . . . . . . . . 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 ) ) ) )
8685ex 434 . . . . . . . 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 ) ) ) ) )
87 3mix3 1167 . . . . . . . . . 10  |-  ( ( y  =  (/)  /\  x  =  2o )  ->  (
( y  =  1o 
/\  x  =  (/) )  \/  ( y  =  1o  /\  x  =  2o )  \/  (
y  =  (/)  /\  x  =  2o ) ) )
88873mix3d 1173 . . . . . . . . 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 ) ) ) )
8988expcom 435 . . . . . . . 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 ) ) ) ) )
9083, 86, 893jaod 1292 . . . . . . 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 ) ) ) ) )
91 3mix1 1165 . . . . . . . . . 10  |-  ( ( y  =  1o  /\  x  =  (/) )  -> 
( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
92913mix3d 1173 . . . . . . . . 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 ) ) ) )
9392expcom 435 . . . . . . . 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 ) ) ) ) )
94 3mix3 1167 . . . . . . . . . 10  |-  ( ( x  =  (/)  /\  y  =  2o )  ->  (
( x  =  1o 
/\  y  =  (/) )  \/  ( x  =  1o  /\  y  =  2o )  \/  (
x  =  (/)  /\  y  =  2o ) ) )
95943mix1d 1171 . . . . . . . . 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 ) ) ) )
9695ex 434 . . . . . . . 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 ) ) ) ) )
97 eqtr3 2485 . . . . . . . . . 10  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  x  =  y )
98973mix2d 1172 . . . . . . . . 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 ) ) ) )
9998ex 434 . . . . . . . 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 ) ) ) ) )
10093, 96, 993jaod 1292 . . . . . . 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 ) ) ) ) )
10180, 90, 1003jaoi 1291 . . . . . 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 ) ) ) ) )
102101imp 429 . . . . 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 ) ) ) )
10369, 70, 102syl2anb 479 . . . 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 ) ) ) )
104 biid 236 . . . . 5  |-  ( x  =  y  <->  x  =  y )
10528, 24brtp 29353 . . . . 5  |-  ( y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x  <->  ( ( y  =  1o  /\  x  =  (/) )  \/  (
y  =  1o  /\  x  =  2o )  \/  ( y  =  (/)  /\  x  =  2o ) ) )
10629, 104, 1053orbi123i 1186 . . . 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 ) ) ) )
107103, 106sylibr 212 . . 3  |-  ( ( x  e.  { 1o ,  2o ,  (/) }  /\  y  e.  { 1o ,  2o ,  (/) } )  ->  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. } y  \/  x  =  y  \/  y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } x ) )
10827, 68, 107issoi 4840 . 2  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  { 1o ,  2o ,  (/) }
109 df-tp 4037 . . 3  |-  { 1o ,  2o ,  (/) }  =  ( { 1o ,  2o }  u.  { (/) } )
110 soeq2 4829 . . 3  |-  ( { 1o ,  2o ,  (/)
}  =  ( { 1o ,  2o }  u.  { (/) } )  -> 
( { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  { 1o ,  2o ,  (/) }  <->  { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } ) ) )
111109, 110ax-mp 5 . 2  |-  ( {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  Or  { 1o ,  2o ,  (/)
}  <->  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
} ) )
112108, 111mpbi 208 1  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819    u. cun 3469   (/)c0 3793   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456    Or wor 4808   Oncon0 4887   suc csuc 4889   1oc1o 7141   2oc2o 7142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-1o 7148  df-2o 7149
This theorem is referenced by:  sltso  29603
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