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Theorem unifpw 7815
Description: A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
unifpw  |-  U. ( ~P A  i^i  Fin )  =  A

Proof of Theorem unifpw
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 inss1 3704 . . . . . 6  |-  ( ~P A  i^i  Fin )  C_ 
~P A
21unissi 4258 . . . . 5  |-  U. ( ~P A  i^i  Fin )  C_ 
U. ~P A
3 unipw 4687 . . . . 5  |-  U. ~P A  =  A
42, 3sseqtri 3521 . . . 4  |-  U. ( ~P A  i^i  Fin )  C_  A
54sseli 3485 . . 3  |-  ( a  e.  U. ( ~P A  i^i  Fin )  ->  a  e.  A )
6 snelpwi 4682 . . . . . 6  |-  ( a  e.  A  ->  { a }  e.  ~P A
)
7 snfi 7589 . . . . . . 7  |-  { a }  e.  Fin
87a1i 11 . . . . . 6  |-  ( a  e.  A  ->  { a }  e.  Fin )
96, 8elind 3674 . . . . 5  |-  ( a  e.  A  ->  { a }  e.  ( ~P A  i^i  Fin )
)
10 elssuni 4264 . . . . 5  |-  ( { a }  e.  ( ~P A  i^i  Fin )  ->  { a } 
C_  U. ( ~P A  i^i  Fin ) )
119, 10syl 16 . . . 4  |-  ( a  e.  A  ->  { a }  C_  U. ( ~P A  i^i  Fin )
)
12 snidg 4042 . . . 4  |-  ( a  e.  A  ->  a  e.  { a } )
1311, 12sseldd 3490 . . 3  |-  ( a  e.  A  ->  a  e.  U. ( ~P A  i^i  Fin ) )
145, 13impbii 188 . 2  |-  ( a  e.  U. ( ~P A  i^i  Fin )  <->  a  e.  A )
1514eqriv 2450 1  |-  U. ( ~P A  i^i  Fin )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   {csn 4016   U.cuni 4235   Fincfn 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-om 6674  df-1o 7122  df-en 7510  df-fin 7513
This theorem is referenced by:  isacs5lem  16001  acsmapd  16010  acsmap2d  16011
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