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Theorem inf1 7368
Description: Variation of Axiom of Infinity (using zfinf 7385 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 3495 . . . 4  |-  ( y  e.  x  ->  x  =/=  (/) )
32anim1i 551 . . 3  |-  ( ( 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
) ) ) )
43eximi 1567 . 2  |-  ( E. x ( y  e.  x  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) )  ->  E. x ( x  =/=  (/)  /\  A. y
( y  e.  x  ->  E. z ( y  e.  z  /\  z  e.  x ) ) ) )
51, 4ax-mp 8 1  |-  E. x
( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
( y  e.  z  /\  z  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531   E.wex 1532    e. wcel 1701    =/= wne 2479   (/)c0 3489
This theorem is referenced by:  inf2  7369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-v 2824  df-dif 3189  df-nul 3490
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