Theorem omex 9090

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 9068.

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 7571 and Fin = V (the universe of all sets) by fineqv 8717. 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 7581 through peano5 7585 (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 9089	. 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 3125	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
4		peano5 7585	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) → ω ⊆ 𝑥)
5	3, 4	sylan2 595	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ω ⊆ 𝑥)
6	5	eximi 1836	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∃𝑥ω ⊆ 𝑥)
7		vex 3444	. . . 4 ⊢ 𝑥 ∈ V
8	7	ssex 5189	. . 3 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
9	8	exlimiv 1931	. 2 ⊢ (∃𝑥ω ⊆ 𝑥 → ω ∈ V)
10	1, 6, 9	mp2b 10	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  suc csuc 6161  ωcom 7560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-om 7561

This theorem is referenced by:  axinf  9091  inf5  9092  omelon  9093  dfom3  9094  elom3  9095  oancom  9098  isfinite  9099  nnsdom  9101  omenps  9102  omensuc  9103  unbnn3  9106  noinfep  9107  tz9.1  9155  tz9.1c  9156  xpct  9427  fseqdom  9437  fseqen  9438  aleph0  9477  alephprc  9510  alephfplem1  9515  alephfplem4  9518  iunfictbso  9525  unctb  9616  r1om  9655  cfom  9675  itunifval  9827  hsmexlem5  9841  axcc2lem  9847  acncc  9851  axcc4dom  9852  domtriomlem  9853  axdclem2  9931  fnct  9948  infinf  9977  unirnfdomd  9978  alephval2  9983  dominfac  9984  iunctb  9985  pwfseqlem4  10073  pwfseqlem5  10074  pwxpndom2  10076  pwdjundom  10078  gchac  10092  wunex2  10149  tskinf  10180  niex  10292  nnexALT  11627  ltweuz  13324  uzenom  13327  nnenom  13343  axdc4uzlem  13346  seqex  13366  rexpen  15573  cctop  21611  2ndcctbss  22060  2ndcdisj  22061  2ndcdisj2  22062  tx2ndc  22256  met2ndci  23129  snct  30475  bnj852  32303  bnj865  32305  satf  32713  satom  32716  satfv0  32718  satfvsuclem1  32719  satfv1lem  32722  satf00  32734  satf0suclem  32735  satf0suc  32736  sat1el2xp  32739  fmla  32741  fmlasuc0  32744  ex-sategoelel  32781  ex-sategoelelomsuc  32786  ex-sategoelel12  32787  prv1n  32791  trpredex  33189  bj-iomnnom  34674  iunctb2  34820  ctbssinf  34823