Theorem omex 9331

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 9309.

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 7699 and Fin = V (the universe of all sets) by fineqv 8967. 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 7710 through peano5 7714 (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 9330	. 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 3083	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
4		peano5 7714	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) → ω ⊆ 𝑥)
5	3, 4	sylan2 592	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ω ⊆ 𝑥)
6	5	eximi 1838	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∃𝑥ω ⊆ 𝑥)
7		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
8	7	ssex 5240	. . 3 ⊢ (ω ⊆ 𝑥 → ω ∈ V)
9	8	exlimiv 1934	. 2 ⊢ (∃𝑥ω ⊆ 𝑥 → ω ∈ V)
10	1, 6, 9	mp2b 10	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  suc csuc 6253  ωcom 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-om 7688

This theorem is referenced by:  axinf  9332  inf5  9333  omelon  9334  dfom3  9335  elom3  9336  oancom  9339  isfinite  9340  nnsdom  9342  omenps  9343  omensuc  9344  unbnn3  9347  noinfep  9348  trpredex  9416  tz9.1  9418  tz9.1c  9419  xpct  9703  fseqdom  9713  fseqen  9714  aleph0  9753  alephprc  9786  alephfplem1  9791  alephfplem4  9794  iunfictbso  9801  unctb  9892  r1om  9931  cfom  9951  itunifval  10103  hsmexlem5  10117  axcc2lem  10123  acncc  10127  axcc4dom  10128  domtriomlem  10129  axdclem2  10207  fnct  10224  infinf  10253  unirnfdomd  10254  alephval2  10259  dominfac  10260  iunctb  10261  pwfseqlem4  10349  pwfseqlem5  10350  pwxpndom2  10352  pwdjundom  10354  gchac  10368  wunex2  10425  tskinf  10456  niex  10568  nnexALT  11905  ltweuz  13609  uzenom  13612  nnenom  13628  axdc4uzlem  13631  seqex  13651  rexpen  15865  cctop  22064  2ndcctbss  22514  2ndcdisj  22515  2ndcdisj2  22516  tx2ndc  22710  met2ndci  23584  snct  30950  bnj852  32801  bnj865  32803  satf  33215  satom  33218  satfv0  33220  satfvsuclem1  33221  satfv1lem  33224  satf00  33236  satf0suclem  33237  satf0suc  33238  sat1el2xp  33241  fmla  33243  fmlasuc0  33246  ex-sategoelel  33283  ex-sategoelelomsuc  33288  ex-sategoelel12  33289  prv1n  33293  ttrclse  33713  bj-iomnnom  35357  iunctb2  35501  ctbssinf  35504