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Theorem en2eleq 27381
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 6638 . . . . . 6  |-  2o  e.  om
2 nnfi 7053 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 8 . . . . 5  |-  2o  e.  Fin
4 enfi 7079 . . . . 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 6637 . . . . . . . . 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 6480 . . . . . . . . 9  |-  2o  =  suc  1o
1210, 11syl6breq 4062 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  suc  1o )
13 dif1en 7091 . . . . . . . 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 27380 . . . . . . 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 3749 . . . . . 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 3771 . . . 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 2529 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  =/=  U. ( P 
\  { X }
) )
24 pr2nelem 7634 . . . . 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 6910 . . . . 5  |-  ( P 
~~  2o  ->  2o  ~~  P )
2726adantl 452 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  2o  ~~  P )
28 entr 6913 . . . 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 7074 . . 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 2288 1  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641   U.cuni 3827   class class class wbr 4023   suc csuc 4394   omcom 4656   1oc1o 6472   2oc2o 6473    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  en2other2  27382  psgnunilem1  27416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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