Theorem zfinf2 4503

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

Step	Hyp	Ref	Expression
1		ax-iinf 4502	. 2 |- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
2		df-ral 2421	. . . 4 |- ( A. y e. x suc y e. x <-> A. y ( y e. x -> suc y e. x ) )
3	2	anbi2i 452	. . 3 |- ( ( (/) e. x /\ A. y e. x suc y e. x ) <-> ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) ) )
4	3	exbii 1584	. 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 ) ) )
5	1, 4	mpbir 145	1 |- E. x ( (/) e. x /\ A. y e. x suc y e. x )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468   e. wcel 1480   A.wral 2416   (/)c0 3363   suc csuc 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-ral 2421

This theorem is referenced by:  omex  4507