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Theorem omex 7360
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 7338.

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 4683 and  Fin  =  _V (the universe of all sets) by fineqv 7094. 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 4691 through peano5 4695 (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 7359 . 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 2628 . . . 4  |-  ( A. y  e.  x  suc  y  e.  x  ->  A. y  e.  om  (
y  e.  x  ->  suc  y  e.  x
) )
4 peano5 4695 . . . 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 1566 . 2  |-  ( E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
)  ->  E. x om  C_  x )
7 vex 2804 . . . 4  |-  x  e. 
_V
87ssex 4174 . . 3  |-  ( om  C_  x  ->  om  e.  _V )
98exlimiv 1624 . 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 1531    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   suc csuc 4410   omcom 4672
This theorem is referenced by:  axinf  7361  inf5  7362  omelon  7363  dfom3  7364  elom3  7365  oancom  7368  isfinite  7369  nnsdom  7370  omenps  7371  omensuc  7372  unbnn3  7375  noinfep  7376  noinfepOLD  7377  tz9.1  7427  tz9.1c  7428  fseqdom  7669  fseqen  7670  aleph0  7709  alephprc  7742  alephfplem1  7747  alephfplem4  7750  iunfictbso  7757  unctb  7847  r1om  7886  cfom  7906  itunifval  8058  hsmexlem5  8072  axcc2lem  8078  acncc  8082  axcc4dom  8083  domtriomlem  8084  axdclem2  8163  infinf  8204  unirnfdomd  8205  alephval2  8210  dominfac  8211  iunctb  8212  pwfseqlem4  8300  pwfseqlem5  8301  pwxpndom2  8303  pwcdandom  8305  gchac  8311  wunex2  8376  tskinf  8407  niex  8521  nnexALT  9764  ltweuz  11040  uzenom  11043  nnenom  11058  axdc4uzlem  11060  seqex  11064  rexpen  12522  cctop  16759  2ndcctbss  17197  2ndcdisj  17198  2ndcdisj2  17199  tx1stc  17360  tx2ndc  17361  met2ndci  18084  xpct  23353  snct  23354  fnct  23356  trpredex  24310  trclval  25996  cartarlim  26007  bnj852  29268  bnj865  29270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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