Theorem omex 9555

Description: The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. Remark 1.21 of [Schloeder] p. 3. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 9533.

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 7822 and Fin = V (the universe of all sets) by fineqv 9170. 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 7833 through peano5 7837 (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.

Step	Hyp	Ref	Expression
1		vex 3434	. . 3 ⊢ 𝑥 ∈ V
2	1	ssex 5258	. 2 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
3		zfinf2 9554	. . 3 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
4		ax-1 6	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) → (𝑦 ∈ ω → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)))
5	4	ralimi2 3070	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
6		peano5 7837	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) → ω ⊆ 𝑥)
7	5, 6	sylan2 594	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ω ⊆ 𝑥)
8	3, 7	eximii 1839	. 2 ⊢ ∃𝑥ω ⊆ 𝑥
9	2, 8	exlimiiv 1933	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  suc csuc 6319  ωcom 7810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-om 7811

This theorem is referenced by:  axinf  9556  inf5  9557  omelon  9558  dfom3  9559  elom3  9560  oancom  9563  isfinite  9564  nnsdom  9566  omenps  9567  omensuc  9568  unbnn3  9571  noinfep  9572  ttrclse  9639  tz9.1  9641  tz9.1c  9642  xpct  9929  fseqdom  9939  fseqen  9940  aleph0  9979  alephprc  10012  alephfplem1  10017  alephfplem4  10020  iunfictbso  10027  unctb  10117  r1om  10156  cfom  10177  itunifval  10329  hsmexlem5  10343  axcc2lem  10349  acncc  10353  axcc4dom  10354  domtriomlem  10355  axdclem2  10433  fnct  10450  infinf  10480  unirnfdomd  10481  alephval2  10486  dominfac  10487  iunctb  10488  pwfseqlem4  10576  pwfseqlem5  10577  pwxpndom2  10579  pwdjundom  10581  gchac  10595  wunex2  10652  tskinf  10683  niex  10795  nnexALT  12167  ltweuz  13914  uzenom  13917  nnenom  13933  axdc4uzlem  13936  seqex  13956  rexpen  16186  cctop  22981  2ndcctbss  23430  2ndcdisj  23431  2ndcdisj2  23432  tx2ndc  23626  met2ndci  24497  n0sex  28323  n0ssold  28360  snct  32800  bnj852  35079  bnj865  35081  r1omfv  35270  satf  35551  satom  35554  satfv0  35556  satfvsuclem1  35557  satfv1lem  35560  satf00  35572  satf0suclem  35573  satf0suc  35574  sat1el2xp  35577  fmla  35579  fmlasuc0  35582  ex-sategoelel  35619  ex-sategoelelomsuc  35624  ex-sategoelel12  35625  prv1n  35629  bj-iomnnom  37589  iunctb2  37733  ctbssinf  37736  succlg  43774  finonex  43899  orbitex  45400