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Theorem inf1 7931
Description: Variation of Axiom of Infinity (using zfinf 7948 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
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
inf1  |-  E. x
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )

Proof of Theorem inf1
StepHypRef Expression
1 inf1.1 . 2  |-  E. x
( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
2 ne0i 3743 . . 3  |-  ( y  e.  x  ->  x  =/=  (/) )
32anim1i 568 . 2  |-  ( ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )  -> 
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) ) )
41, 3eximii 1628 1  |-  E. x
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587    =/= wne 2644   (/)c0 3737
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  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-v 3072  df-dif 3431  df-nul 3738
This theorem is referenced by:  inf2  7932
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