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Theorem zfinf2 4340
Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
Assertion
Ref Expression
zfinf2  |-  E. x
( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
Distinct variable group:    x, y

Proof of Theorem zfinf2
StepHypRef Expression
1 ax-iinf 4339 . 2  |-  E. x
( (/)  e.  x  /\  A. y ( y  e.  x  ->  suc  y  e.  x ) )
2 df-ral 2328 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  <->  A. y
( y  e.  x  ->  suc  y  e.  x
) )
32anbi2i 438 . . 3  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  <->  (
(/)  e.  x  /\  A. y ( y  e.  x  ->  suc  y  e.  x ) ) )
43exbii 1512 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  <->  E. x ( (/)  e.  x  /\  A. y
( y  e.  x  ->  suc  y  e.  x
) ) )
51, 4mpbir 138 1  |-  E. x
( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257   E.wex 1397    e. wcel 1409   A.wral 2323   (/)c0 3252   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  omex  4344
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