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Theorem ssfin2 8696
 Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
ssfin2 FinII FinII

Proof of Theorem ssfin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . 4 FinII FinII
2 elpwi 4019 . . . . . 6
32adantl 466 . . . . 5 FinII
4 simplr 754 . . . . . 6 FinII
5 sspwb 4696 . . . . . 6
64, 5sylib 196 . . . . 5 FinII
73, 6sstrd 3514 . . . 4 FinII
8 fin2i 8671 . . . . 5 FinII []
98ex 434 . . . 4 FinII []
101, 7, 9syl2anc 661 . . 3 FinII []
1110ralrimiva 2878 . 2 FinII []
12 ssexg 4593 . . . 4 FinII
1312ancoms 453 . . 3 FinII
14 isfin2 8670 . . 3 FinII []
1513, 14syl 16 . 2 FinII FinII []
1611, 15mpbird 232 1 FinII FinII
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wcel 1767   wne 2662  wral 2814  cvv 3113   wss 3476  c0 3785  cpw 4010  cuni 4245   wor 4799   [] crpss 6561  FinIIcfin2 8655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-uni 4246  df-po 4800  df-so 4801  df-fin2 8662 This theorem is referenced by: (None)
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