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Theorem en2eqpr 8419
Description: Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
en2eqpr  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )

Proof of Theorem en2eqpr
StepHypRef Expression
1 2onn 7328 . . . . . 6  |-  2o  e.  om
2 nnfi 7750 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 simpl1 1002 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  ~~  2o )
5 enfii 7774 . . . . 5  |-  ( ( 2o  e.  Fin  /\  C  ~~  2o )  ->  C  e.  Fin )
63, 4, 5sylancr 663 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  e.  Fin )
7 simpl2 1003 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  A  e.  C )
8 simpl3 1004 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  B  e.  C )
9 prssi 4130 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
107, 8, 9syl2anc 661 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  C_  C
)
11 pr2nelem 8416 . . . . . . 7  |-  ( ( A  e.  C  /\  B  e.  C  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
12113expa 1199 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
13123adantl1 1155 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  2o )
144ensymd 7606 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  2o  ~~  C
)
15 entr 7607 . . . . 5  |-  ( ( { A ,  B }  ~~  2o  /\  2o  ~~  C )  ->  { A ,  B }  ~~  C
)
1613, 14, 15syl2anc 661 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  C
)
17 fisseneq 7768 . . . 4  |-  ( ( C  e.  Fin  /\  { A ,  B }  C_  C  /\  { A ,  B }  ~~  C
)  ->  { A ,  B }  =  C )
186, 10, 16, 17syl3anc 1232 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  =  C )
1918eqcomd 2412 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
2019ex 434 1  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600    C_ wss 3416   {cpr 3976   class class class wbr 4397   omcom 6685   2oc2o 7163    ~~ cen 7553   Fincfn 7556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-1o 7169  df-2o 7170  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560
This theorem is referenced by:  en2top  19781
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