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Theorem inf2 7570
Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7586 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 7569 . 2  |-  E. x
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
3 dfss2 3329 . . . . 5  |-  ( x 
C_  U. x  <->  A. y
( y  e.  x  ->  y  e.  U. x
) )
4 eluni 4010 . . . . . . 7  |-  ( y  e.  U. x  <->  E. z
( y  e.  z  /\  z  e.  x
) )
54imbi2i 304 . . . . . 6  |-  ( ( y  e.  x  -> 
y  e.  U. x
)  <->  ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
65albii 1575 . . . . 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 241 . . . 4  |-  ( x 
C_  U. x  <->  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) )
87anbi2i 676 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  <->  ( x  =/=  (/)  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
98exbii 1592 . 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 201 1  |-  E. x
( x  =/=  (/)  /\  x  C_ 
U. x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   U.cuni 4007
This theorem is referenced by:  axinf2  7587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-uni 4008
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