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Theorem inf2 7319
Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7335 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
Hypothesis
Ref Expression
inf1.1  |-  E. x
( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
Assertion
Ref Expression
inf2  |-  E. x
( x  =/=  (/)  /\  x  C_ 
U. x )
Distinct variable group:    x, y, z

Proof of Theorem inf2
StepHypRef Expression
1 inf1.1 . . 3  |-  E. x
( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
21inf1 7318 . 2  |-  E. x
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
3 dfss2 3170 . . . . 5  |-  ( x 
C_  U. x  <->  A. y
( y  e.  x  ->  y  e.  U. x
) )
4 eluni 3831 . . . . . . 7  |-  ( y  e.  U. x  <->  E. z
( y  e.  z  /\  z  e.  x
) )
54imbi2i 305 . . . . . 6  |-  ( ( y  e.  x  -> 
y  e.  U. x
)  <->  ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
65albii 1558 . . . . 5  |-  ( A. y ( y  e.  x  ->  y  e.  U. x )  <->  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) )
73, 6bitri 242 . . . 4  |-  ( x 
C_  U. x  <->  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) )
87anbi2i 678 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  <->  ( x  =/=  (/)  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
98exbii 1574 . 2  |-  ( E. x ( x  =/=  (/)  /\  x  C_  U. x
)  <->  E. x ( x  =/=  (/)  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
102, 9mpbir 202 1  |-  E. x
( x  =/=  (/)  /\  x  C_ 
U. x )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1533    e. wcel 1688    =/= wne 2447    C_ wss 3153   (/)c0 3456   U.cuni 3828
This theorem is referenced by:  axinf2  7336
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-in 3160  df-ss 3167  df-nul 3457  df-uni 3829
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