Theorem omex 3475

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. A finitist (someone who doesn't believe in infinity) could, without contradiction, omit the axiom of Infinity and instead deny it; this would lead to ω = On (the proper class of ordinals) by omon 2384 and onprc 2240. 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 2390 through peano5 2394 (which many textbooks prove more easily assuming Infinity). The mathematics used by computers is essentially finitist; for example, computers cannot work directly with real numbers but only approximations of them in the form of floating-point numbers.

Assertion

Ref	Expression
omex	⊢ ω ∈ V

Proof of Theorem omex

Step	Hyp	Ref	Expression
1		zfinf 3474	. . 3 ⊢ ∃x(∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)
2		peano5 2394	. . . . 5 ⊢ ((∅ ∈ x ∧ ∀y ∈ ω (y ∈ x → suc y ∈ x)) → ω ⊆ x)
3		ax-1 3	. . . . . 6 ⊢ ((y ∈ x → suc y ∈ x) → (y ∈ ω → (y ∈ x → suc y ∈ x)))
4	3	r19.20i2 1252	. . . . 5 ⊢ (∀y ∈ x suc y ∈ x → ∀y ∈ ω (y ∈ x → suc y ∈ x))
5	2, 4	sylan2 346	. . . 4 ⊢ ((∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) → ω ⊆ x)
6	5	19.22i 723	. . 3 ⊢ (∃x(∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) → ∃xω ⊆ x)
7	1, 6	ax-mp 6	. 2 ⊢ ∃xω ⊆ x
8		visset 1350	. . . 4 ⊢ x ∈ V
9	8	ssex 1700	. . 3 ⊢ (ω ⊆ x → ω ∈ V)
10	9	19.23aiv 952	. 2 ⊢ (∃xω ⊆ x → ω ∈ V)
11	7, 10	ax-mp 6	1 ⊢ ω ∈ V

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wel 803   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  suc csuc 2201  ωcom 2372

This theorem is referenced by:  omelon 3476  dfom3 3477  elom3 3478  isfinite 3480  nnsdom 3481  omenps 3482  omensuc 3483  tz9.1 3490  sucdom 3648  aleph0 3669  alephprc 3698  cfom 3710  cdainf 3731  niex 3803  nnenom 4926  xpomen 4928  infxpidmlem10 4942  infdif 4948

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970

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