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Definition df-fin2 8698
Description: A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
Assertion
Ref Expression
df-fin2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) }
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-fin2
StepHypRef Expression
1 cfin2 8691 . 2  class FinII
2 vy . . . . . . . 8  setvar  y
32cv 1404 . . . . . . 7  class  y
4 c0 3738 . . . . . . 7  class  (/)
53, 4wne 2598 . . . . . 6  wff  y  =/=  (/)
6 crpss 6561 . . . . . . 7  class [ C.]
73, 6wor 4743 . . . . . 6  wff [ C.]  Or  y
85, 7wa 367 . . . . 5  wff  ( y  =/=  (/)  /\ [ C.]  Or  y
)
93cuni 4191 . . . . . 6  class  U. y
109, 3wcel 1842 . . . . 5  wff  U. y  e.  y
118, 10wi 4 . . . 4  wff  ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )
12 vx . . . . . . 7  setvar  x
1312cv 1404 . . . . . 6  class  x
1413cpw 3955 . . . . 5  class  ~P x
1514cpw 3955 . . . 4  class  ~P ~P x
1611, 2, 15wral 2754 . . 3  wff  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )
1716, 12cab 2387 . 2  class  { x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) }
181, 17wceq 1405 1  wff FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) }
Colors of variables: wff setvar class
This definition is referenced by:  isfin2  8706
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