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Theorem zfinf2 4255
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 4254 . 2 x(∅ x y(y x → suc y x))
2 df-ral 2305 . . . 4 (y x suc y xy(y x → suc y x))
32anbi2i 430 . . 3 ((∅ x y x suc y x) ↔ (∅ x y(y x → suc y x)))
43exbii 1493 . 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 1240  wex 1378   wcel 1390  wral 2300  c0 3218  suc csuc 4068
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-ral 2305
This theorem is referenced by:  omex  4259
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