ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  zfinf2 Structured version   GIF version

Theorem zfinf2 4239
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 x(∅ x y x suc y x)
Distinct variable group:   x,y

Proof of Theorem zfinf2
StepHypRef Expression
1 ax-iinf 4238 . 2 x(∅ x y(y x → suc y x))
2 df-ral 2289 . . . 4 (y x suc y xy(y x → suc y x))
32anbi2i 433 . . 3 ((∅ x y x suc y x) ↔ (∅ x y(y x → suc y x)))
43exbii 1478 . 2 (x(∅ x y x suc y x) ↔ x(∅ x y(y x → suc y x)))
51, 4mpbir 134 1 x(∅ x y x suc y x)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226  wex 1362   wcel 1374  wral 2284  c0 3201  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-ral 2289
This theorem is referenced by:  omex  4243
  Copyright terms: Public domain W3C validator