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Theorem omex 7590
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 7568.

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 4848 and  Fin  =  _V (the universe of all sets) by fineqv 7316. 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 4856 through peano5 4860 (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 7589 . 2  |-  E. x
( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
2 ax-1 5 . . . . 5  |-  ( ( y  e.  x  ->  suc  y  e.  x
)  ->  ( y  e.  om  ->  ( y  e.  x  ->  suc  y  e.  x ) ) )
32ralimi2 2770 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  ->  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )
4 peano5 4860 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )  ->  om  C_  x
)
53, 4sylan2 461 . . 3  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  ->  om  C_  x )
65eximi 1585 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  ->  E. x om  C_  x )
7 vex 2951 . . . 4  |-  x  e. 
_V
87ssex 4339 . . 3  |-  ( om  C_  x  ->  om  e.  _V )
98exlimiv 1644 . 2  |-  ( E. x om  C_  x  ->  om  e.  _V )
101, 6, 9mp2b 10 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   (/)c0 3620   suc csuc 4575   omcom 4837
This theorem is referenced by:  axinf  7591  inf5  7592  omelon  7593  dfom3  7594  elom3  7595  oancom  7598  isfinite  7599  nnsdom  7600  omenps  7601  omensuc  7602  unbnn3  7605  noinfep  7606  noinfepOLD  7607  tz9.1  7657  tz9.1c  7658  fseqdom  7899  fseqen  7900  aleph0  7939  alephprc  7972  alephfplem1  7977  alephfplem4  7980  iunfictbso  7987  unctb  8077  r1om  8116  cfom  8136  itunifval  8288  hsmexlem5  8302  axcc2lem  8308  acncc  8312  axcc4dom  8313  domtriomlem  8314  axdclem2  8392  infinf  8433  unirnfdomd  8434  alephval2  8439  dominfac  8440  iunctb  8441  pwfseqlem4  8529  pwfseqlem5  8530  pwxpndom2  8532  pwcdandom  8534  gchac  8540  wunex2  8605  tskinf  8636  niex  8750  nnexALT  9994  ltweuz  11293  uzenom  11296  nnenom  11311  axdc4uzlem  11313  seqex  11317  rexpen  12819  cctop  17062  2ndcctbss  17510  2ndcdisj  17511  2ndcdisj2  17512  tx1stc  17674  tx2ndc  17675  met2ndci  18544  xpct  24094  snct  24095  fnct  24097  trpredex  25507  bnj852  29229  bnj865  29231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838
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