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Theorem omex 8484
Description: The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 8462.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 7023 and Fin = V (the universe of all sets) by fineqv 8119. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7032 through peano5 7036 (which many textbooks prove more easily assuming Infinity). (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 8483 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
2 ax-1 6 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) → (𝑦 ∈ ω → (𝑦𝑥 → suc 𝑦𝑥)))
32ralimi2 2944 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 → ∀𝑦 ∈ ω (𝑦𝑥 → suc 𝑦𝑥))
4 peano5 7036 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦𝑥 → suc 𝑦𝑥)) → ω ⊆ 𝑥)
53, 4sylan2 491 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ω ⊆ 𝑥)
65eximi 1759 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∃𝑥ω ⊆ 𝑥)
7 vex 3189 . . . 4 𝑥 ∈ V
87ssex 4762 . . 3 (ω ⊆ 𝑥 → ω ∈ V)
98exlimiv 1855 . 2 (∃𝑥ω ⊆ 𝑥 → ω ∈ V)
101, 6, 9mp2b 10 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wcel 1987  wral 2907  Vcvv 3186  wss 3555  c0 3891  suc csuc 5684  ωcom 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-om 7013
This theorem is referenced by:  axinf  8485  inf5  8486  omelon  8487  dfom3  8488  elom3  8489  oancom  8492  isfinite  8493  nnsdom  8495  omenps  8496  omensuc  8497  unbnn3  8500  noinfep  8501  tz9.1  8549  tz9.1c  8550  xpct  8783  fseqdom  8793  fseqen  8794  aleph0  8833  alephprc  8866  alephfplem1  8871  alephfplem4  8874  iunfictbso  8881  unctb  8971  r1om  9010  cfom  9030  itunifval  9182  hsmexlem5  9196  axcc2lem  9202  acncc  9206  axcc4dom  9207  domtriomlem  9208  axdclem2  9286  fnct  9303  infinf  9332  unirnfdomd  9333  alephval2  9338  dominfac  9339  iunctb  9340  pwfseqlem4  9428  pwfseqlem5  9429  pwxpndom2  9431  pwcdandom  9433  gchac  9447  wunex2  9504  tskinf  9535  niex  9647  nnexALT  10966  ltweuz  12700  uzenom  12703  nnenom  12719  axdc4uzlem  12722  seqex  12743  rexpen  14882  cctop  20720  2ndcctbss  21168  2ndcdisj  21169  2ndcdisj2  21170  tx1stc  21363  tx2ndc  21364  met2ndci  22237  snct  29331  bnj852  30696  bnj865  30698  trpredex  31435
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