Theorem zfinf2 4330

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 4329	. 2 |- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
2		df-ral 2353	. . . 4 |- ( A. y e. x suc y e. x <-> A. y ( y e. x -> suc y e. x ) )
3	2	anbi2i 444	. . 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 1536	. 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 144	1 |- E. x ( (/) e. x /\ A. y e. x suc y e. x )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   e. wcel 1433   A.wral 2348   (/)c0 3251   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  omex  4334