ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2oconcl Unicode version

Theorem 2oconcl 6266
Description: Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
2oconcl  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )

Proof of Theorem 2oconcl
StepHypRef Expression
1 elpri 3497 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 difeq2 3135 . . . . . . . 8  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  ( 1o  \  (/) ) )
3 dif0 3380 . . . . . . . 8  |-  ( 1o 
\  (/) )  =  1o
42, 3syl6eq 2148 . . . . . . 7  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  1o )
5 difeq2 3135 . . . . . . . 8  |-  ( A  =  1o  ->  ( 1o  \  A )  =  ( 1o  \  1o ) )
6 difid 3378 . . . . . . . 8  |-  ( 1o 
\  1o )  =  (/)
75, 6syl6eq 2148 . . . . . . 7  |-  ( A  =  1o  ->  ( 1o  \  A )  =  (/) )
84, 7orim12i 717 . . . . . 6  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  1o  \/  ( 1o  \  A )  =  (/) ) )
98orcomd 689 . . . . 5  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  (/)  \/  ( 1o  \  A )  =  1o ) )
101, 9syl 14 . . . 4  |-  ( A  e.  { (/) ,  1o }  ->  ( ( 1o 
\  A )  =  (/)  \/  ( 1o  \  A )  =  1o ) )
11 1on 6250 . . . . . 6  |-  1o  e.  On
12 difexg 4009 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  \  A )  e. 
_V )
1311, 12ax-mp 7 . . . . 5  |-  ( 1o 
\  A )  e. 
_V
1413elpr 3495 . . . 4  |-  ( ( 1o  \  A )  e.  { (/) ,  1o } 
<->  ( ( 1o  \  A )  =  (/)  \/  ( 1o  \  A
)  =  1o ) )
1510, 14sylibr 133 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  { (/)
,  1o } )
16 df2o3 6257 . . 3  |-  2o  =  { (/) ,  1o }
1715, 16syl6eleqr 2193 . 2  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  2o )
1817, 16eleq2s 2194 1  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 670    = wceq 1299    e. wcel 1448   _Vcvv 2641    \ cdif 3018   (/)c0 3310   {cpr 3475   Oncon0 4223   1oc1o 6236   2oc2o 6237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231  df-1o 6243  df-2o 6244
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator