Theorem omex 9564

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 9542.

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 7829 and Fin = V (the universe of all sets) by fineqv 9177. 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 7840 through peano5 7844 (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 3433	. . 3 ⊢ 𝑥 ∈ V
2	1	ssex 5262	. 2 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
3		zfinf2 9563	. . 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 3069	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
6		peano5 7844	. . . 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 3051  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  suc csuc 6325  ωcom 7817

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-om 7818

This theorem is referenced by:  axinf  9565  inf5  9566  omelon  9567  dfom3  9568  elom3  9569  oancom  9572  isfinite  9573  nnsdom  9575  omenps  9576  omensuc  9577  unbnn3  9580  noinfep  9581  ttrclse  9648  tz9.1  9650  tz9.1c  9651  xpct  9938  fseqdom  9948  fseqen  9949  aleph0  9988  alephprc  10021  alephfplem1  10026  alephfplem4  10029  iunfictbso  10036  unctb  10126  r1om  10165  cfom  10186  itunifval  10338  hsmexlem5  10352  axcc2lem  10358  acncc  10362  axcc4dom  10363  domtriomlem  10364  axdclem2  10442  fnct  10459  infinf  10489  unirnfdomd  10490  alephval2  10495  dominfac  10496  iunctb  10497  pwfseqlem4  10585  pwfseqlem5  10586  pwxpndom2  10588  pwdjundom  10590  gchac  10604  wunex2  10661  tskinf  10692  niex  10804  nnexALT  12176  ltweuz  13923  uzenom  13926  nnenom  13942  axdc4uzlem  13945  seqex  13965  rexpen  16195  cctop  22971  2ndcctbss  23420  2ndcdisj  23421  2ndcdisj2  23422  tx2ndc  23616  met2ndci  24487  n0sex  28309  n0ssold  28346  snct  32785  bnj852  35063  bnj865  35065  r1omfv  35254  satf  35535  satom  35538  satfv0  35540  satfvsuclem1  35541  satfv1lem  35544  satf00  35556  satf0suclem  35557  satf0suc  35558  sat1el2xp  35561  fmla  35563  fmlasuc0  35566  ex-sategoelel  35603  ex-sategoelelomsuc  35608  ex-sategoelel12  35609  prv1n  35613  bj-iomnnom  37573  iunctb2  37719  ctbssinf  37722  succlg  43756  finonex  43881  orbitex  45382