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Theorem axsltsolem1 23725
Description: Lemma for axsltso 23726. The sign expansion relationship totally orders the surreal signs. (Contributed by axsltsolem1, 8-Jun-2011.)
Assertion
Ref Expression
axsltsolem1  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/) } )

Proof of Theorem axsltsolem1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 6490 . . . . . . . 8  |-  1o  =/=  (/)
2 df-ne 2449 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
31, 2mpbi 199 . . . . . . 7  |-  -.  1o  =  (/)
4 eqtr2 2302 . . . . . . 7  |-  ( ( x  =  1o  /\  x  =  (/) )  ->  1o  =  (/) )
53, 4mto 167 . . . . . 6  |-  -.  (
x  =  1o  /\  x  =  (/) )
6 1on 6482 . . . . . . . . 9  |-  1o  e.  On
7 0elon 4444 . . . . . . . . 9  |-  (/)  e.  On
8 df-2o 6476 . . . . . . . . . . 11  |-  2o  =  suc  1o
9 df-1o 6475 . . . . . . . . . . 11  |-  1o  =  suc  (/)
108, 9eqeq12i 2297 . . . . . . . . . 10  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
11 suc11 4495 . . . . . . . . . 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 2535 . . . . . . 7  |-  -.  2o  =  1o
15 eqtr2 2302 . . . . . . . 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 4471 . . . . . . . 8  |-  suc  1o  =/=  (/)
198eqeq1i 2291 . . . . . . . 8  |-  ( 2o  =  (/)  <->  suc  1o  =  (/) )
2018, 19nemtbir 2535 . . . . . . 7  |-  -.  2o  =  (/)
21 eqtr2 2302 . . . . . . . 8  |-  ( ( x  =  2o  /\  x  =  (/) )  ->  2o  =  (/) )
2221ancoms 439 . . . . . . 7  |-  ( ( x  =  (/)  /\  x  =  2o )  ->  2o  =  (/) )
2320, 22mto 167 . . . . . 6  |-  -.  (
x  =  (/)  /\  x  =  2o )
245, 17, 233pm3.2ni 23471 . . . . 5  |-  -.  (
( x  =  1o 
/\  x  =  (/) )  \/  ( x  =  1o  /\  x  =  2o )  \/  (
x  =  (/)  /\  x  =  2o ) )
25 vex 2792 . . . . . 6  |-  x  e. 
_V
2625, 25brtp 23512 . . . . 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 2792 . . . . . . 7  |-  y  e. 
_V
3025, 29brtp 23512 . . . . . 6  |-  ( x { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } y  <->  ( ( x  =  1o  /\  y  =  (/) )  \/  (
x  =  1o  /\  y  =  2o )  \/  ( x  =  (/)  /\  y  =  2o ) ) )
31 vex 2792 . . . . . . 7  |-  z  e. 
_V
3229, 31brtp 23512 . . . . . 6  |-  ( y { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. } z  <->  ( ( y  =  1o  /\  z  =  (/) )  \/  (
y  =  1o  /\  z  =  2o )  \/  ( y  =  (/)  /\  z  =  2o ) ) )
33 eqtr2 2302 . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . 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 23512 . . . . 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 3679 . . . . 5  |-  ( x  e.  { 1o ,  2o ,  (/) }  <->  ( x  =  1o  \/  x  =  2o  \/  x  =  (/) ) )
7129eltp 3679 . . . . 5  |-  ( y  e.  { 1o ,  2o ,  (/) }  <->  ( y  =  1o  \/  y  =  2o  \/  y  =  (/) ) )
72 eqtr3 2303 . . . . . . . . . 10  |-  ( ( x  =  1o  /\  y  =  1o )  ->  x  =  y )
73723mix2d 23475 . . . . . . . . 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 23474 . . . . . . . . 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 23474 . . . . . . . . 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 23476 . . . . . . . . 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 2303 . . . . . . . . . 10  |-  ( ( x  =  2o  /\  y  =  2o )  ->  x  =  y )
86853mix2d 23475 . . . . . . . . 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 23476 . . . . . . . . 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 23476 . . . . . . . . 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 23474 . . . . . . . . 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 2303 . . . . . . . . . 10  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  x  =  y )
99983mix2d 23475 . . . . . . . . 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 23512 . . . . 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 4344 . 2  |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  { 1o ,  2o ,  (/) }
110 df-tp 3649 . . 3  |-  { 1o ,  2o ,  (/) }  =  ( { 1o ,  2o }  u.  { (/) } )
111 soeq2 4333 . . 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 1623    e. wcel 1685    =/= wne 2447    u. cun 3151   (/)c0 3456   {csn 3641   {cpr 3642   {ctp 3643   <.cop 3644   class class class wbr 4024    Or wor 4312   Oncon0 4391   suc csuc 4393   1oc1o 6468   2oc2o 6469
This theorem is referenced by:  axsltso  23726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-1o 6475  df-2o 6476
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