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Theorem en2eleq 8187
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 7091 . . . . . 6  |-  2o  e.  om
2 nnfi 7515 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 enfi 7541 . . . . 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 7090 . . . . . . . . 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 6933 . . . . . . . . 9  |-  2o  =  suc  1o
1210, 11syl6breq 4343 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  suc  1o )
13 dif1en 7557 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  X  e.  P )  ->  ( P  \  { X } )  ~~  1o )
149, 12, 7, 13syl3anc 1218 . . . . . . 7  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( P  \  { X } )  ~~  1o )
15 en1uniel 7393 . . . . . . 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 4012 . . . . . 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 4041 . . . 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 2707 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  =/=  U. ( P 
\  { X }
) )
24 pr2nelem 8183 . . . . 5  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P  /\  X  =/=  U. ( P  \  { X }
) )  ->  { X ,  U. ( P  \  { X } ) } 
~~  2o )
257, 19, 23, 24syl3anc 1218 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  2o )
26 ensym 7370 . . . . 5  |-  ( P 
~~  2o  ->  2o  ~~  P )
2726adantl 466 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  2o  ~~  P )
28 entr 7373 . . . 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 7536 . . 3  |-  ( ( P  e.  Fin  /\  { X ,  U. ( P  \  { X }
) }  C_  P  /\  { X ,  U. ( P  \  { X } ) }  ~~  P )  ->  { X ,  U. ( P  \  { X } ) }  =  P )
316, 21, 29, 30syl3anc 1218 . 2  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  =  P )
3231eqcomd 2448 1  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618    \ cdif 3337    C_ wss 3340   {csn 3889   {cpr 3891   U.cuni 4103   class class class wbr 4304   suc csuc 4733   omcom 6488   1oc1o 6925   2oc2o 6926    ~~ cen 7319   Fincfn 7322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-1o 6932  df-2o 6933  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326
This theorem is referenced by:  en2other2  8188  psgnunilem1  16011
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