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Theorem 2oconcl 6433
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 3614 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 difeq2 3247 . . . . . . . 8  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  ( 1o  \  (/) ) )
3 dif0 3493 . . . . . . . 8  |-  ( 1o 
\  (/) )  =  1o
42, 3eqtrdi 2226 . . . . . . 7  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  1o )
5 difeq2 3247 . . . . . . . 8  |-  ( A  =  1o  ->  ( 1o  \  A )  =  ( 1o  \  1o ) )
6 difid 3491 . . . . . . . 8  |-  ( 1o 
\  1o )  =  (/)
75, 6eqtrdi 2226 . . . . . . 7  |-  ( A  =  1o  ->  ( 1o  \  A )  =  (/) )
84, 7orim12i 759 . . . . . 6  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  1o  \/  ( 1o  \  A )  =  (/) ) )
98orcomd 729 . . . . 5  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  (/)  \/  ( 1o  \  A )  =  1o ) )
101, 9syl 14 . . . 4  |-  ( A  e.  { (/) ,  1o }  ->  ( ( 1o 
\  A )  =  (/)  \/  ( 1o  \  A )  =  1o ) )
11 1on 6417 . . . . . 6  |-  1o  e.  On
12 difexg 4141 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  \  A )  e. 
_V )
1311, 12ax-mp 5 . . . . 5  |-  ( 1o 
\  A )  e. 
_V
1413elpr 3612 . . . 4  |-  ( ( 1o  \  A )  e.  { (/) ,  1o } 
<->  ( ( 1o  \  A )  =  (/)  \/  ( 1o  \  A
)  =  1o ) )
1510, 14sylibr 134 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  { (/)
,  1o } )
16 df2o3 6424 . . 3  |-  2o  =  { (/) ,  1o }
1715, 16eleqtrrdi 2271 . 2  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  2o )
1817, 16eleq2s 2272 1  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 708    = wceq 1353    e. wcel 2148   _Vcvv 2737    \ cdif 3126   (/)c0 3422   {cpr 3592   Oncon0 4359   1oc1o 6403   2oc2o 6404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-iord 4362  df-on 4364  df-suc 4367  df-1o 6410  df-2o 6411
This theorem is referenced by:  ismkvnex  7146
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