MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2oconcl Structured version   Unicode version

Theorem 2oconcl 6939
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 3894 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 difeq2 3465 . . . . . . . 8  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  ( 1o  \  (/) ) )
3 dif0 3746 . . . . . . . 8  |-  ( 1o 
\  (/) )  =  1o
42, 3syl6eq 2489 . . . . . . 7  |-  ( A  =  (/)  ->  ( 1o 
\  A )  =  1o )
5 difeq2 3465 . . . . . . . 8  |-  ( A  =  1o  ->  ( 1o  \  A )  =  ( 1o  \  1o ) )
6 difid 3744 . . . . . . . 8  |-  ( 1o 
\  1o )  =  (/)
75, 6syl6eq 2489 . . . . . . 7  |-  ( A  =  1o  ->  ( 1o  \  A )  =  (/) )
84, 7orim12i 513 . . . . . 6  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  1o  \/  ( 1o  \  A )  =  (/) ) )
98orcomd 388 . . . . 5  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  (
( 1o  \  A
)  =  (/)  \/  ( 1o  \  A )  =  1o ) )
101, 9syl 16 . . . 4  |-  ( A  e.  { (/) ,  1o }  ->  ( ( 1o 
\  A )  =  (/)  \/  ( 1o  \  A )  =  1o ) )
11 1on 6923 . . . . . 6  |-  1o  e.  On
12 difexg 4437 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  \  A )  e. 
_V )
1311, 12ax-mp 5 . . . . 5  |-  ( 1o 
\  A )  e. 
_V
1413elpr 3892 . . . 4  |-  ( ( 1o  \  A )  e.  { (/) ,  1o } 
<->  ( ( 1o  \  A )  =  (/)  \/  ( 1o  \  A
)  =  1o ) )
1510, 14sylibr 212 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  { (/)
,  1o } )
16 df2o3 6929 . . 3  |-  2o  =  { (/) ,  1o }
1715, 16syl6eleqr 2532 . 2  |-  ( A  e.  { (/) ,  1o }  ->  ( 1o  \  A )  e.  2o )
1817, 16eleq2s 2533 1  |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   (/)c0 3634   {cpr 3876   Oncon0 4715   1oc1o 6909   2oc2o 6910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-1o 6916  df-2o 6917
This theorem is referenced by:  efgmf  16203  efgmnvl  16204  efglem  16206  frgpuplem  16262
  Copyright terms: Public domain W3C validator