Theorem omex 9680

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 9658.

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 7898 and Fin = V (the universe of all sets) by fineqv 9296. 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 7910 through peano5 7915 (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		zfinf2 9679	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
2		ax-1 6	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) → (𝑦 ∈ ω → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)))
3	2	ralimi2 3075	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
4		peano5 7915	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) → ω ⊆ 𝑥)
5	3, 4	sylan2 593	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ω ⊆ 𝑥)
6	5	eximi 1831	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∃𝑥ω ⊆ 𝑥)
7		vex 3481	. . . 4 ⊢ 𝑥 ∈ V
8	7	ssex 5326	. . 3 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
9	8	exlimiv 1927	. 2 ⊢ (∃𝑥ω ⊆ 𝑥 → ω ∈ V)
10	1, 6, 9	mp2b 10	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1775   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ⊆ wss 3962  ∅c0 4338  suc csuc 6387  ωcom 7886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-inf2 9678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-om 7887

This theorem is referenced by:  axinf  9681  inf5  9682  omelon  9683  dfom3  9684  elom3  9685  oancom  9688  isfinite  9689  nnsdom  9691  omenps  9692  omensuc  9693  unbnn3  9696  noinfep  9697  ttrclse  9764  tz9.1  9766  tz9.1c  9767  xpct  10053  fseqdom  10063  fseqen  10064  aleph0  10103  alephprc  10136  alephfplem1  10141  alephfplem4  10144  iunfictbso  10151  unctb  10241  r1om  10280  cfom  10301  itunifval  10453  hsmexlem5  10467  axcc2lem  10473  acncc  10477  axcc4dom  10478  domtriomlem  10479  axdclem2  10557  fnct  10574  infinf  10603  unirnfdomd  10604  alephval2  10609  dominfac  10610  iunctb  10611  pwfseqlem4  10699  pwfseqlem5  10700  pwxpndom2  10702  pwdjundom  10704  gchac  10718  wunex2  10775  tskinf  10806  niex  10918  nnexALT  12265  ltweuz  13998  uzenom  14001  nnenom  14017  axdc4uzlem  14020  seqex  14040  rexpen  16260  cctop  23028  2ndcctbss  23478  2ndcdisj  23479  2ndcdisj2  23480  tx2ndc  23674  met2ndci  24550  snct  32730  bnj852  34913  bnj865  34915  satf  35337  satom  35340  satfv0  35342  satfvsuclem1  35343  satfv1lem  35346  satf00  35358  satf0suclem  35359  satf0suc  35360  sat1el2xp  35363  fmla  35365  fmlasuc0  35368  ex-sategoelel  35405  ex-sategoelelomsuc  35410  ex-sategoelel12  35411  prv1n  35415  bj-iomnnom  37241  iunctb2  37385  ctbssinf  37388  succlg  43317  finonex  43443