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Theorem omex 7848
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 7826.

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 by omon 6486 and  Fin  =  _V (the universe of all sets) by fineqv 7527. 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 6494 through peano5 6498 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

Assertion
Ref Expression
omex  |-  om  e.  _V

Proof of Theorem omex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfinf2 7847 . 2  |-  E. x
( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
2 ax-1 6 . . . . 5  |-  ( ( y  e.  x  ->  suc  y  e.  x
)  ->  ( y  e.  om  ->  ( y  e.  x  ->  suc  y  e.  x ) ) )
32ralimi2 2787 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  ->  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )
4 peano5 6498 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )  ->  om  C_  x
)
53, 4sylan2 474 . . 3  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  ->  om  C_  x )
65eximi 1625 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  ->  E. x om  C_  x )
7 vex 2974 . . . 4  |-  x  e. 
_V
87ssex 4435 . . 3  |-  ( om  C_  x  ->  om  e.  _V )
98exlimiv 1688 . 2  |-  ( E. x om  C_  x  ->  om  e.  _V )
101, 6, 9mp2b 10 1  |-  om  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1586    e. wcel 1756   A.wral 2714   _Vcvv 2971    C_ wss 3327   (/)c0 3636   suc csuc 4720   omcom 6475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371  ax-inf2 7846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-om 6476
This theorem is referenced by:  axinf  7849  inf5  7850  omelon  7851  dfom3  7852  elom3  7853  oancom  7856  isfinite  7857  nnsdom  7858  omenps  7859  omensuc  7860  unbnn3  7863  noinfep  7864  noinfepOLD  7865  tz9.1  7948  tz9.1c  7949  fseqdom  8195  fseqen  8196  aleph0  8235  alephprc  8268  alephfplem1  8273  alephfplem4  8276  iunfictbso  8283  unctb  8373  r1om  8412  cfom  8432  itunifval  8584  hsmexlem5  8598  axcc2lem  8604  acncc  8608  axcc4dom  8609  domtriomlem  8610  axdclem2  8688  infinf  8729  unirnfdomd  8730  alephval2  8735  dominfac  8736  iunctb  8737  pwfseqlem4  8828  pwfseqlem5  8829  pwxpndom2  8831  pwcdandom  8833  gchac  8847  wunex2  8904  tskinf  8935  niex  9049  nnexALT  10323  ltweuz  11783  uzenom  11786  nnenom  11801  axdc4uzlem  11803  seqex  11807  rexpen  13509  cctop  18609  2ndcctbss  19058  2ndcdisj  19059  2ndcdisj2  19060  tx1stc  19222  tx2ndc  19223  met2ndci  20096  xpct  26009  snct  26010  fnct  26012  trpredex  27700  bnj852  31912  bnj865  31914
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