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Theorem omex 7339
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 7317.

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 4666 and  Fin  =  _V (the universe of all sets) by fineqv 7073. 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 4674 through peano5 4678 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

Assertion
Ref Expression
omex  |-  om  e.  _V
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem omex
StepHypRef Expression
1 zfinf2 7338 . 2  |-  E. x
( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
2 ax-1 7 . . . . 5  |-  ( ( y  e.  x  ->  suc  y  e.  x
)  ->  ( y  e.  om  ->  ( y  e.  x  ->  suc  y  e.  x ) ) )
32ralimi2 2616 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  ->  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )
4 peano5 4678 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )  ->  om  C_  x
)
53, 4sylan2 462 . . 3  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  ->  om  C_  x )
65eximi 1564 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  ->  E. x om  C_  x )
7 vex 2792 . . . 4  |-  x  e. 
_V
87ssex 4159 . . 3  |-  ( om  C_  x  ->  om  e.  _V )
98exlimiv 1667 . 2  |-  ( E. x om  C_  x  ->  om  e.  _V )
101, 6, 9mp2b 11 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1529    e. wcel 1685   A.wral 2544   _Vcvv 2789    C_ wss 3153   (/)c0 3456   suc csuc 4393   omcom 4655
This theorem is referenced by:  axinf  7340  inf5  7341  omelon  7342  dfom3  7343  elom3  7344  oancom  7347  isfinite  7348  nnsdom  7349  omenps  7350  omensuc  7351  unbnn3  7354  noinfep  7355  noinfepOLD  7356  tz9.1  7406  tz9.1c  7407  fseqdom  7648  fseqen  7649  aleph0  7688  alephprc  7721  alephfplem1  7726  alephfplem4  7729  iunfictbso  7736  unctb  7826  r1om  7865  cfom  7885  itunifval  8037  hsmexlem5  8051  axcc2lem  8057  acncc  8061  axcc4dom  8062  domtriomlem  8063  axdclem2  8142  infinf  8183  unirnfdomd  8184  alephval2  8189  dominfac  8190  iunctb  8191  pwfseqlem4  8279  pwfseqlem5  8280  pwxpndom2  8282  pwcdandom  8284  gchac  8290  wunex2  8355  tskinf  8386  niex  8500  nnexALT  9743  ltweuz  11018  uzenom  11021  nnenom  11036  axdc4uzlem  11038  seqex  11042  rexpen  12500  cctop  16737  2ndcctbss  17175  2ndcdisj  17176  2ndcdisj2  17177  tx1stc  17338  tx2ndc  17339  met2ndci  18062  trpredex  23641  trclval  25293  cartarlim  25304  bnj852  28220  bnj865  28222
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656
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