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Theorem 2oconcl 6544
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 3694 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 difeq2 3322 . . . . . . . 8  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  ( 1o  \  (/) ) )
3 dif0 3558 . . . . . . . 8  |-  ( 1o 
\  (/) )  =  1o
42, 3syl6eq 2364 . . . . . . 7  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  1o )
5 difeq2 3322 . . . . . . . 8  |-  ( A  =  1o  ->  ( 1o  \  A )  =  ( 1o  \  1o ) )
6 difid 3556 . . . . . . . 8  |-  ( 1o 
\  1o )  =  (/)
75, 6syl6eq 2364 . . . . . . 7  |-  ( A  =  1o  ->  ( 1o  \  A )  =  (/) )
84, 7orim12i 502 . . . . . 6  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  1o  \/  ( 1o  \  A )  =  (/) ) )
98orcomd 377 . . . . 5  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  (/)  \/  ( 1o  \  A )  =  1o ) )
101, 9syl 15 . . . 4  |-  ( A  e.  { (/) ,  1o }  ->  ( ( 1o 
\  A )  =  (/)  \/  ( 1o  \  A )  =  1o ) )
11 1on 6528 . . . . . 6  |-  1o  e.  On
12 difexg 4199 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  \  A )  e. 
_V )
1311, 12ax-mp 8 . . . . 5  |-  ( 1o 
\  A )  e. 
_V
1413elpr 3692 . . . 4  |-  ( ( 1o  \  A )  e.  { (/) ,  1o } 
<->  ( ( 1o  \  A )  =  (/)  \/  ( 1o  \  A
)  =  1o ) )
1510, 14sylibr 203 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  { (/)
,  1o } )
16 df2o3 6534 . . 3  |-  2o  =  { (/) ,  1o }
1715, 16syl6eleqr 2407 . 2  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  2o )
1817, 16eleq2s 2408 1  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183   (/)c0 3489   {cpr 3675   Oncon0 4429   1oc1o 6514   2oc2o 6515
This theorem is referenced by:  efgmf  15071  efgmnvl  15072  efglem  15074  frgpuplem  15130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-1o 6521  df-2o 6522
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