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Theorem fin2i 8206
 Description: Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin2i FinII []

Proof of Theorem fin2i
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pwexg 4412 . . . . 5 FinII
2 elpw2g 4392 . . . . 5
31, 2syl 16 . . . 4 FinII
43biimpar 473 . . 3 FinII
5 isfin2 8205 . . . . 5 FinII FinII []
65ibi 234 . . . 4 FinII []
76adantr 453 . . 3 FinII []
8 neeq1 2615 . . . . . 6
9 soeq2 4552 . . . . . 6 [] []
108, 9anbi12d 693 . . . . 5 [] []
11 unieq 4048 . . . . . 6
12 id 21 . . . . . 6
1311, 12eleq12d 2510 . . . . 5
1410, 13imbi12d 313 . . . 4 [] []
1514rspcv 3054 . . 3 [] []
164, 7, 15sylc 59 . 2 FinII []
1716imp 420 1 FinII []
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1727   wne 2605  wral 2711  cvv 2962   wss 3306  c0 3613  cpw 3823  cuni 4039   wor 4531   [] crpss 6550  FinIIcfin2 8190 This theorem is referenced by:  fin2i2  8229  ssfin2  8231  enfin2i  8232  fin1a2lem13  8323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-pow 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-v 2964  df-in 3313  df-ss 3320  df-pw 3825  df-uni 4040  df-po 4532  df-so 4533  df-fin2 8197
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