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Theorem omex 7344
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 7322.

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 4667 and  Fin  =  _V (the universe of all sets) by fineqv 7078. 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 4675 through peano5 4679 (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 7343 . 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 2615 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  ->  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )
4 peano5 4679 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )  ->  om  C_  x
)
53, 4sylan2 460 . . 3  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  ->  om  C_  x )
65eximi 1563 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  ->  E. x om  C_  x )
7 vex 2791 . . . 4  |-  x  e. 
_V
87ssex 4158 . . 3  |-  ( om  C_  x  ->  om  e.  _V )
98exlimiv 1666 . 2  |-  ( E. x om  C_  x  ->  om  e.  _V )
101, 6, 9mp2b 9 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   suc csuc 4394   omcom 4656
This theorem is referenced by:  axinf  7345  inf5  7346  omelon  7347  dfom3  7348  elom3  7349  oancom  7352  isfinite  7353  nnsdom  7354  omenps  7355  omensuc  7356  unbnn3  7359  noinfep  7360  noinfepOLD  7361  tz9.1  7411  tz9.1c  7412  fseqdom  7653  fseqen  7654  aleph0  7693  alephprc  7726  alephfplem1  7731  alephfplem4  7734  iunfictbso  7741  unctb  7831  r1om  7870  cfom  7890  itunifval  8042  hsmexlem5  8056  axcc2lem  8062  acncc  8066  axcc4dom  8067  domtriomlem  8068  axdclem2  8147  infinf  8188  unirnfdomd  8189  alephval2  8194  dominfac  8195  iunctb  8196  pwfseqlem4  8284  pwfseqlem5  8285  pwxpndom2  8287  pwcdandom  8289  gchac  8295  wunex2  8360  tskinf  8391  niex  8505  nnexALT  9748  ltweuz  11024  uzenom  11027  nnenom  11042  axdc4uzlem  11044  seqex  11048  rexpen  12506  cctop  16743  2ndcctbss  17181  2ndcdisj  17182  2ndcdisj2  17183  tx1stc  17344  tx2ndc  17345  met2ndci  18068  xpct  23338  snct  23339  fnct  23341  trpredex  24240  trclval  25894  cartarlim  25905  bnj852  28953  bnj865  28955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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