Theorem omex 4607

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 4586.

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 (the proper class of ordinals) by omon 3138 and onprc 2984. 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 3144 through peano5 3148 (which many textbooks prove more easily assuming Infinity).

Assertion

Ref	Expression
omex	⊢ ω ∈ V

Proof of Theorem omex

Step	Hyp	Ref	Expression
1		zfinf 4606	. . 3 ⊢ ∃x(∅ ∈ x ⋀ ∀y ∈ x suc y ∈ x)
2		peano5 3148	. . . . 5 ⊢ ((∅ ∈ x ⋀ ∀y ∈ ω (y ∈ x → suc y ∈ x)) → ω ⊆ x)
3		ax-1 4	. . . . . 6 ⊢ ((y ∈ x → suc y ∈ x) → (y ∈ ω → (y ∈ x → suc y ∈ x)))
4	3	r19.20i2 1700	. . . . 5 ⊢ (∀y ∈ x suc y ∈ x → ∀y ∈ ω (y ∈ x → suc y ∈ x))
5	2, 4	sylan2 451	. . . 4 ⊢ ((∅ ∈ x ⋀ ∀y ∈ x suc y ∈ x) → ω ⊆ x)
6	5	19.22i 1038	. . 3 ⊢ (∃x(∅ ∈ x ⋀ ∀y ∈ x suc y ∈ x) → ∃xω ⊆ x)
7	1, 6	ax-mp 7	. 2 ⊢ ∃xω ⊆ x
8		visset 1809	. . . 4 ⊢ x ∈ V
9	8	ssex 2714	. . 3 ⊢ (ω ⊆ x → ω ∈ V)
10	9	19.23aiv 1293	. 2 ⊢ (∃xω ⊆ x → ω ∈ V)
11	7, 10	ax-mp 7	1 ⊢ ω ∈ V

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 956  ∃wex 978  ∀wral 1642  Vcvv 1807   ⊆ wss 2043  ∅c0 2276  suc csuc 2945  ωcom 3126

This theorem is referenced by:  inf5 4608  omelon 4609  dfom3 4610  elom3 4611  oancom 4613  isfinite 4614  nnsdom 4615  omenps 4616  omensuc 4617  unbnnt 4619  noinfep 4620  tz9.1 4626  sucdom 4822  aleph0 4843  alephprc 4873  alephfplem4 4879  alephval2 4882  dominf 4884  cfom 4896  cdainf 4917  niex 4989  nnenom 7448  xpomen 7450  unben 7456  aleph1re 7502  infxpidmlem10 7512  infdif 7519  iunctb 7525  aleph1irr 7528

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127

Copyright terms: Public domain