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Theorem omex 4259
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 4255 . . 3 y(∅ y x y suc x y)
2 intexabim 3897 . . 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 4258 . . 3 𝜔 = {y ∣ (∅ y x y suc x y)}
54eleq1i 2100 . 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 1378   wcel 1390  {cab 2023  wral 2300  Vcvv 2551  c0 3218   cint 3606  suc csuc 4068  𝜔com 4256
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-iom 4257
This theorem is referenced by:  peano5  4264  omelon  4274  frecabex  5923  niex  6296  enq0ex  6421  nq0ex  6422  uzenom  8843  frecfzennn  8844  nnenom  8851
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