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Theorem omex 4344
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 4340 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)
2 intexabim 3934 . . 3 (∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ∈ V)
31, 2ax-mp 7 . 2 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ∈ V
4 dfom3 4343 . . 3 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
54eleq1i 2119 . 2 (ω ∈ V ↔ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ∈ V)
63, 5mpbir 138 1 ω ∈ V
Colors of variables: wff set class
Syntax hints:  wa 101  wex 1397  wcel 1409  {cab 2042  wral 2323  Vcvv 2574  c0 3252   cint 3643  suc csuc 4130  ωcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-int 3644  df-iom 4342
This theorem is referenced by:  peano5  4349  omelon  4359  frecex  6012  frecabex  6015  niex  6468  enq0ex  6595  nq0ex  6596  uzenom  9366  frecfzennn  9367  nnenom  9374
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