Theorem omex 4627

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 4606.

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 3143 and onprc 2989. 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 3149 through peano5 3153 (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 4626	. . 3 |- E.x((/) e. x /\ A.y e. x suc y e. x)
2		peano5 3153	. . . . 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 1703	. . . . 5 |- (A.y e. x suc y e. x -> A.y e. om (y e. x -> suc y e. x))
5	2, 4	sylan2 451	. . . 4 |- (((/) e. x /\ A.y e. x suc y e. x) -> om (_ x)
6	5	19.22i 1040	. . 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 1813	. . . 4 |- x e. V
9	8	ssex 2719	. . 3 |- (om (_ x -> om e. V)
10	9	19.23aiv 1295	. 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 958  E.wex 980  A.wral 1645  Vcvv 1811   (_ wss 2047  (/)c0 2280  suc csuc 2950  omcom 3131

This theorem is referenced by:  inf5 4628  omelon 4629  dfom3 4630  elom3 4631  oancom 4633  isfiniteOLD 4634  nnsdom 4635  omenps 4636  omensuc 4637  unbnnt 4639  noinfep 4640  tz9.1 4646  sucdom 4842  aleph0 4863  alephprc 4893  alephfplem4 4899  alephval2 4902  dominf 4904  cfom 4916  cdainf 4937  niex 5009  nnenom 7498  xpomen 7500  unben 7505  aleph1re 7551  infxpidmlem10 7561  infdif 7568  iunctb 7575  aleph1irr 7578

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132

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