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Theorem 2oconcl 6342
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 3553 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 difeq2 3191 . . . . . . . 8  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  ( 1o  \  (/) ) )
3 dif0 3436 . . . . . . . 8  |-  ( 1o 
\  (/) )  =  1o
42, 3eqtrdi 2189 . . . . . . 7  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  1o )
5 difeq2 3191 . . . . . . . 8  |-  ( A  =  1o  ->  ( 1o  \  A )  =  ( 1o  \  1o ) )
6 difid 3434 . . . . . . . 8  |-  ( 1o 
\  1o )  =  (/)
75, 6eqtrdi 2189 . . . . . . 7  |-  ( A  =  1o  ->  ( 1o  \  A )  =  (/) )
84, 7orim12i 749 . . . . . 6  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  1o  \/  ( 1o  \  A )  =  (/) ) )
98orcomd 719 . . . . 5  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  (/)  \/  ( 1o  \  A )  =  1o ) )
101, 9syl 14 . . . 4  |-  ( A  e.  { (/) ,  1o }  ->  ( ( 1o 
\  A )  =  (/)  \/  ( 1o  \  A )  =  1o ) )
11 1on 6326 . . . . . 6  |-  1o  e.  On
12 difexg 4075 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  \  A )  e. 
_V )
1311, 12ax-mp 5 . . . . 5  |-  ( 1o 
\  A )  e. 
_V
1413elpr 3551 . . . 4  |-  ( ( 1o  \  A )  e.  { (/) ,  1o } 
<->  ( ( 1o  \  A )  =  (/)  \/  ( 1o  \  A
)  =  1o ) )
1510, 14sylibr 133 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  { (/)
,  1o } )
16 df2o3 6333 . . 3  |-  2o  =  { (/) ,  1o }
1715, 16eleqtrrdi 2234 . 2  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  2o )
1817, 16eleq2s 2235 1  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 698    = wceq 1332    e. wcel 1481   _Vcvv 2689    \ cdif 3071   (/)c0 3366   {cpr 3531   Oncon0 4291   1oc1o 6312   2oc2o 6313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-uni 3743  df-tr 4033  df-iord 4294  df-on 4296  df-suc 4299  df-1o 6319  df-2o 6320
This theorem is referenced by:  ismkvnex  7035
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