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Theorem en2eleq 8387
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 7290 . . . . . 6  |-  2o  e.  om
2 nnfi 7711 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 enfi 7737 . . . . 5  |-  ( P 
~~  2o  ->  ( P  e.  Fin  <->  2o  e.  Fin ) )
53, 4mpbiri 233 . . . 4  |-  ( P 
~~  2o  ->  P  e. 
Fin )
65adantl 466 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  e.  Fin )
7 simpl 457 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  e.  P )
8 1onn 7289 . . . . . . . . 9  |-  1o  e.  om
98a1i 11 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  1o  e.  om )
10 simpr 461 . . . . . . . . 9  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  2o )
11 df-2o 7132 . . . . . . . . 9  |-  2o  =  suc  1o
1210, 11syl6breq 4486 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  suc  1o )
13 dif1en 7754 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  X  e.  P )  ->  ( P  \  { X } )  ~~  1o )
149, 12, 7, 13syl3anc 1228 . . . . . . 7  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( P  \  { X } )  ~~  1o )
15 en1uniel 7588 . . . . . . 7  |-  ( ( P  \  { X } )  ~~  1o  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
1614, 15syl 16 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
17 eldifsn 4152 . . . . . 6  |-  ( U. ( P  \  { X } )  e.  ( P  \  { X } )  <->  ( U. ( P  \  { X } )  e.  P  /\  U. ( P  \  { X } )  =/= 
X ) )
1816, 17sylib 196 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( U. ( P 
\  { X }
)  e.  P  /\  U. ( P  \  { X } )  =/=  X
) )
1918simpld 459 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  P
)
20 prssi 4183 . . . 4  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P
)  ->  { X ,  U. ( P  \  { X } ) } 
C_  P )
217, 19, 20syl2anc 661 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  C_  P
)
2218simprd 463 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  =/=  X
)
2322necomd 2738 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  =/=  U. ( P 
\  { X }
) )
24 pr2nelem 8383 . . . . 5  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P  /\  X  =/=  U. ( P  \  { X }
) )  ->  { X ,  U. ( P  \  { X } ) } 
~~  2o )
257, 19, 23, 24syl3anc 1228 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  2o )
26 ensym 7565 . . . . 5  |-  ( P 
~~  2o  ->  2o  ~~  P )
2726adantl 466 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  2o  ~~  P )
28 entr 7568 . . . 4  |-  ( ( { X ,  U. ( P  \  { X } ) }  ~~  2o  /\  2o  ~~  P
)  ->  { X ,  U. ( P  \  { X } ) } 
~~  P )
2925, 27, 28syl2anc 661 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  P
)
30 fisseneq 7732 . . 3  |-  ( ( P  e.  Fin  /\  { X ,  U. ( P  \  { X }
) }  C_  P  /\  { X ,  U. ( P  \  { X } ) }  ~~  P )  ->  { X ,  U. ( P  \  { X } ) }  =  P )
316, 21, 29, 30syl3anc 1228 . 2  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  =  P )
3231eqcomd 2475 1  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   {csn 4027   {cpr 4029   U.cuni 4245   class class class wbr 4447   suc csuc 4880   omcom 6685   1oc1o 7124   2oc2o 7125    ~~ cen 7514   Fincfn 7517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-1o 7131  df-2o 7132  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521
This theorem is referenced by:  en2other2  8388  psgnunilem1  16333
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