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

Theorem omex 4243
Description: The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
Assertion
Ref Expression
omex 𝜔 V

Proof of Theorem omex
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfinf2 4239 . . 3 y(∅ y x y suc x y)
2 intexabim 3880 . . 3 (y(∅ y x y suc x y) → {y ∣ (∅ y x y suc x y)} V)
31, 2ax-mp 7 . 2 {y ∣ (∅ y x y suc x y)} V
4 dfom3 4242 . . 3 𝜔 = {y ∣ (∅ y x y suc x y)}
54eleq1i 2085 . 2 (𝜔 V ↔ {y ∣ (∅ y x y suc x y)} V)
63, 5mpbir 134 1 𝜔 V
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1362   wcel 1374  {cab 2008  wral 2284  Vcvv 2535  c0 3201   cint 3589  suc csuc 4051  𝜔com 4240
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-int 3590  df-iom 4241
This theorem is referenced by:  peano5  4248  omelon  4258  frecabex  5899  niex  6172  enq0ex  6294  nq0ex  6295
  Copyright terms: Public domain W3C validator