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Theorem omex 8500
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 8478.

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 7038 and Fin = V (the universe of all sets) by fineqv 8135. 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 7047 through peano5 7051 (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 8499 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
2 ax-1 6 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) → (𝑦 ∈ ω → (𝑦𝑥 → suc 𝑦𝑥)))
32ralimi2 2945 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 → ∀𝑦 ∈ ω (𝑦𝑥 → suc 𝑦𝑥))
4 peano5 7051 . . . 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 3193 . . . 4 𝑥 ∈ V
87ssex 4772 . . 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 2908  Vcvv 3190  wss 3560  c0 3897  suc csuc 5694  ωcom 7027
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 4751  ax-nul 4759  ax-pr 4877  ax-un 6914  ax-inf2 8498
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028
This theorem is referenced by:  axinf  8501  inf5  8502  omelon  8503  dfom3  8504  elom3  8505  oancom  8508  isfinite  8509  nnsdom  8511  omenps  8512  omensuc  8513  unbnn3  8516  noinfep  8517  tz9.1  8565  tz9.1c  8566  xpct  8799  fseqdom  8809  fseqen  8810  aleph0  8849  alephprc  8882  alephfplem1  8887  alephfplem4  8890  iunfictbso  8897  unctb  8987  r1om  9026  cfom  9046  itunifval  9198  hsmexlem5  9212  axcc2lem  9218  acncc  9222  axcc4dom  9223  domtriomlem  9224  axdclem2  9302  fnct  9319  infinf  9348  unirnfdomd  9349  alephval2  9354  dominfac  9355  iunctb  9356  pwfseqlem4  9444  pwfseqlem5  9445  pwxpndom2  9447  pwcdandom  9449  gchac  9463  wunex2  9520  tskinf  9551  niex  9663  nnexALT  10982  ltweuz  12716  uzenom  12719  nnenom  12735  axdc4uzlem  12738  seqex  12759  rexpen  14901  cctop  20750  2ndcctbss  21198  2ndcdisj  21199  2ndcdisj2  21200  tx1stc  21393  tx2ndc  21394  met2ndci  22267  snct  29375  bnj852  30752  bnj865  30754  trpredex  31491
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