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Theorem fin2i 8568
 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 4577 . . . . 5 FinII
2 elpw2g 4556 . . . . 5
31, 2syl 16 . . . 4 FinII
43biimpar 485 . . 3 FinII
5 isfin2 8567 . . . . 5 FinII FinII []
65ibi 241 . . . 4 FinII []
76adantr 465 . . 3 FinII []
8 neeq1 2729 . . . . . 6
9 soeq2 4762 . . . . . 6 [] []
108, 9anbi12d 710 . . . . 5 [] []
11 unieq 4200 . . . . . 6
12 id 22 . . . . . 6
1311, 12eleq12d 2533 . . . . 5
1410, 13imbi12d 320 . . . 4 [] []
1514rspcv 3168 . . 3 [] []
164, 7, 15sylc 60 . 2 FinII []
1716imp 429 1 FinII []
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758   wne 2644  wral 2795  cvv 3071   wss 3429  c0 3738  cpw 3961  cuni 4192   wor 4741   [] crpss 6462  FinIIcfin2 8552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-pow 4571 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-pw 3963  df-uni 4193  df-po 4742  df-so 4743  df-fin2 8559 This theorem is referenced by:  fin2i2  8591  ssfin2  8593  enfin2i  8594  fin1a2lem13  8685
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