Theorem omex 4636

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 4615.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial -. om e. V; this would lead to om = On (the proper class of ordinals) by omon 3149 and onprc 2995. 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 3155 through peano5 3159 (which many textbooks prove more easily assuming Infinity).

Assertion

Ref	Expression
omex	|- om e. V

Proof of Theorem omex

Step	Hyp	Ref	Expression
1		zfinf 4635	. . 3 |- E.x((/) e. x /\ A.y e. x suc y e. x)
2		peano5 3159	. . . . 5 |- (((/) e. x /\ A.y e. om (y e. x -> suc y e. x)) -> om (_ x)
3		ax-1 4	. . . . . 6 |- ((y e. x -> suc y e. x) -> (y e. om -> (y e. x -> suc y e. x)))
4	3	r19.20i2 1706	. . . . 5 |- (A.y e. x suc y e. x -> A.y e. om (y e. x -> suc y e. x))
5	2, 4	sylan2 453	. . . 4 |- (((/) e. x /\ A.y e. x suc y e. x) -> om (_ x)
6	5	19.22i 1042	. . 3 |- (E.x((/) e. x /\ A.y e. x suc y e. x) -> E.xom (_ x)
7	1, 6	ax-mp 7	. 2 |- E.xom (_ x
8		visset 1816	. . . 4 |- x e. V
9	8	ssex 2724	. . 3 |- (om (_ x -> om e. V)
10	9	19.23aiv 1297	. 2 |- (E.xom (_ x -> om e. V)
11	7, 10	ax-mp 7	1 |- om e. V

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  E.wex 982  A.wral 1648  Vcvv 1814   (_ wss 2050  (/)c0 2283  suc csuc 2956  omcom 3137

This theorem is referenced by:  inf5 4637  omelon 4638  dfom3 4639  elom3 4640  oancom 4642  isfinite 4643  isfiniteOLD 4644  nnsdom 4645  omenps 4646  omensuc 4647  unbnnt 4649  noinfep 4650  tz9.1 4656  sucdom 4852  sucdomOLD 4853  aleph0 4874  alephprc 4904  alephfplem4 4910  alephval2 4913  dominf 4915  dominfOLD 4916  cfom 4928  cdainf 4949  niex 5021  nnenom 7499  xpomen 7501  unben 7506  aleph1re 7552  infxpidmlem10 7562  infdif 7569  iunctb 7576  aleph1irr 7580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138

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