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Theorem en2eleq 27484
Description: Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Assertion
Ref Expression
en2eleq  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )

Proof of Theorem en2eleq
StepHypRef Expression
1 2onn 6654 . . . . . 6  |-  2o  e.  om
2 nnfi 7069 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 8 . . . . 5  |-  2o  e.  Fin
4 enfi 7095 . . . . 5  |-  ( P 
~~  2o  ->  ( P  e.  Fin  <->  2o  e.  Fin ) )
53, 4mpbiri 224 . . . 4  |-  ( P 
~~  2o  ->  P  e. 
Fin )
65adantl 452 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  e.  Fin )
7 simpl 443 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  e.  P )
8 1onn 6653 . . . . . . . . 9  |-  1o  e.  om
98a1i 10 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  1o  e.  om )
10 simpr 447 . . . . . . . . 9  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  2o )
11 df-2o 6496 . . . . . . . . 9  |-  2o  =  suc  1o
1210, 11syl6breq 4078 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  suc  1o )
13 dif1en 7107 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  X  e.  P )  ->  ( P  \  { X } )  ~~  1o )
149, 12, 7, 13syl3anc 1182 . . . . . . 7  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( P  \  { X } )  ~~  1o )
15 en1uniel 27483 . . . . . . 7  |-  ( ( P  \  { X } )  ~~  1o  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
1614, 15syl 15 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
17 eldifsn 3762 . . . . . 6  |-  ( U. ( P  \  { X } )  e.  ( P  \  { X } )  <->  ( U. ( P  \  { X } )  e.  P  /\  U. ( P  \  { X } )  =/= 
X ) )
1816, 17sylib 188 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( U. ( P 
\  { X }
)  e.  P  /\  U. ( P  \  { X } )  =/=  X
) )
1918simpld 445 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  P
)
20 prssi 3787 . . . 4  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P
)  ->  { X ,  U. ( P  \  { X } ) } 
C_  P )
217, 19, 20syl2anc 642 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  C_  P
)
2218simprd 449 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  =/=  X
)
2322necomd 2542 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  =/=  U. ( P 
\  { X }
) )
24 pr2nelem 7650 . . . . 5  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P  /\  X  =/=  U. ( P  \  { X }
) )  ->  { X ,  U. ( P  \  { X } ) } 
~~  2o )
257, 19, 23, 24syl3anc 1182 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  2o )
26 ensym 6926 . . . . 5  |-  ( P 
~~  2o  ->  2o  ~~  P )
2726adantl 452 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  2o  ~~  P )
28 entr 6929 . . . 4  |-  ( ( { X ,  U. ( P  \  { X } ) }  ~~  2o  /\  2o  ~~  P
)  ->  { X ,  U. ( P  \  { X } ) } 
~~  P )
2925, 27, 28syl2anc 642 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  P
)
30 fisseneq 7090 . . 3  |-  ( ( P  e.  Fin  /\  { X ,  U. ( P  \  { X }
) }  C_  P  /\  { X ,  U. ( P  \  { X } ) }  ~~  P )  ->  { X ,  U. ( P  \  { X } ) }  =  P )
316, 21, 29, 30syl3anc 1182 . 2  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  =  P )
3231eqcomd 2301 1  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   {cpr 3654   U.cuni 3843   class class class wbr 4039   suc csuc 4410   omcom 4672   1oc1o 6488   2oc2o 6489    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  en2other2  27485  psgnunilem1  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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