Theorem omex 8423

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 8401.

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 6968 and Fin = V (the universe of all sets) by fineqv 8060. 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 6977 through peano5 6981 (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 8422	. 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 2933	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
4		peano5 6981	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) → ω ⊆ 𝑥)
5	3, 4	sylan2 490	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ω ⊆ 𝑥)
6	5	eximi 1752	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∃𝑥ω ⊆ 𝑥)
7		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
8	7	ssex 4730	. . 3 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
9	8	exlimiv 1845	. 2 ⊢ (∃𝑥ω ⊆ 𝑥 → ω ∈ V)
10	1, 6, 9	mp2b 10	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  suc csuc 5642  ωcom 6957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958

This theorem is referenced by:  axinf  8424  inf5  8425  omelon  8426  dfom3  8427  elom3  8428  oancom  8431  isfinite  8432  nnsdom  8434  omenps  8435  omensuc  8436  unbnn3  8439  noinfep  8440  tz9.1  8488  tz9.1c  8489  xpct  8722  fseqdom  8732  fseqen  8733  aleph0  8772  alephprc  8805  alephfplem1  8810  alephfplem4  8813  iunfictbso  8820  unctb  8910  r1om  8949  cfom  8969  itunifval  9121  hsmexlem5  9135  axcc2lem  9141  acncc  9145  axcc4dom  9146  domtriomlem  9147  axdclem2  9225  infinf  9267  unirnfdomd  9268  alephval2  9273  dominfac  9274  iunctb  9275  pwfseqlem4  9363  pwfseqlem5  9364  pwxpndom2  9366  pwcdandom  9368  gchac  9382  wunex2  9439  tskinf  9470  niex  9582  nnexALT  10899  ltweuz  12622  uzenom  12625  nnenom  12641  axdc4uzlem  12644  seqex  12665  rexpen  14796  cctop  20620  2ndcctbss  21068  2ndcdisj  21069  2ndcdisj2  21070  tx1stc  21263  tx2ndc  21264  met2ndci  22137  snct  28874  fnct  28876  bnj852  30245  bnj865  30247  trpredex  30981