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Theorem en2eleq 8429
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 7340 . . . . . 6  |-  2o  e.  om
2 nnfi 7762 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 enfi 7785 . . . . 5  |-  ( P 
~~  2o  ->  ( P  e.  Fin  <->  2o  e.  Fin ) )
53, 4mpbiri 236 . . . 4  |-  ( P 
~~  2o  ->  P  e. 
Fin )
65adantl 467 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  e.  Fin )
7 simpl 458 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  e.  P )
8 1onn 7339 . . . . . . . . 9  |-  1o  e.  om
98a1i 11 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  1o  e.  om )
10 simpr 462 . . . . . . . . 9  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  2o )
11 df-2o 7182 . . . . . . . . 9  |-  2o  =  suc  1o
1210, 11syl6breq 4456 . . . . . . . 8  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  ~~  suc  1o )
13 dif1en 7801 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  X  e.  P )  ->  ( P  \  { X } )  ~~  1o )
149, 12, 7, 13syl3anc 1264 . . . . . . 7  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( P  \  { X } )  ~~  1o )
15 en1uniel 7639 . . . . . . 7  |-  ( ( P  \  { X } )  ~~  1o  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
1614, 15syl 17 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  ( P  \  { X } ) )
17 eldifsn 4119 . . . . . 6  |-  ( U. ( P  \  { X } )  e.  ( P  \  { X } )  <->  ( U. ( P  \  { X } )  e.  P  /\  U. ( P  \  { X } )  =/= 
X ) )
1816, 17sylib 199 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  -> 
( U. ( P 
\  { X }
)  e.  P  /\  U. ( P  \  { X } )  =/=  X
) )
1918simpld 460 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  e.  P
)
20 prssi 4150 . . . 4  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P
)  ->  { X ,  U. ( P  \  { X } ) } 
C_  P )
217, 19, 20syl2anc 665 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  C_  P
)
2218simprd 464 . . . . . 6  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { X } )  =/=  X
)
2322necomd 2693 . . . . 5  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  X  =/=  U. ( P 
\  { X }
) )
24 pr2nelem 8425 . . . . 5  |-  ( ( X  e.  P  /\  U. ( P  \  { X } )  e.  P  /\  X  =/=  U. ( P  \  { X }
) )  ->  { X ,  U. ( P  \  { X } ) } 
~~  2o )
257, 19, 23, 24syl3anc 1264 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  2o )
26 ensym 7616 . . . . 5  |-  ( P 
~~  2o  ->  2o  ~~  P )
2726adantl 467 . . . 4  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  2o  ~~  P )
28 entr 7619 . . . 4  |-  ( ( { X ,  U. ( P  \  { X } ) }  ~~  2o  /\  2o  ~~  P
)  ->  { X ,  U. ( P  \  { X } ) } 
~~  P )
2925, 27, 28syl2anc 665 . . 3  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  ~~  P
)
30 fisseneq 7780 . . 3  |-  ( ( P  e.  Fin  /\  { X ,  U. ( P  \  { X }
) }  C_  P  /\  { X ,  U. ( P  \  { X } ) }  ~~  P )  ->  { X ,  U. ( P  \  { X } ) }  =  P )
316, 21, 29, 30syl3anc 1264 . 2  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  { X ,  U. ( P  \  { X }
) }  =  P )
3231eqcomd 2428 1  |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    C_ wss 3433   {csn 3993   {cpr 3995   U.cuni 4213   class class class wbr 4417   suc csuc 5435   omcom 6697   1oc1o 7174   2oc2o 7175    ~~ cen 7565   Fincfn 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-1o 7181  df-2o 7182  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572
This theorem is referenced by:  en2other2  8430  psgnunilem1  17078
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