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Theorem omex 8081
 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 8059. A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ; this would lead to by omon 6711 and (the universe of all sets) by fineqv 7755. 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 6719 through peano5 6723 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)
Assertion
Ref Expression
omex

Proof of Theorem omex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfinf2 8080 . 2
2 ax-1 6 . . . . 5
32ralimi2 2847 . . . 4
4 peano5 6723 . . . 4
53, 4sylan2 474 . . 3
65eximi 1656 . 2
7 vex 3112 . . . 4
87ssex 4596 . . 3
98exlimiv 1722 . 2
101, 6, 9mp2b 10 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  E.wex 1612  e.wcel 1818  A.wral 2807   cvv 3109  C_wss 3475   c0 3784  succsuc 4885   com 6700 This theorem is referenced by:  axinf  8082  inf5  8083  omelon  8084  dfom3  8085  elom3  8086  oancom  8089  isfinite  8090  nnsdom  8091  omenps  8092  omensuc  8093  unbnn3  8096  noinfep  8097  noinfepOLD  8098  tz9.1  8181  tz9.1c  8182  fseqdom  8428  fseqen  8429  aleph0  8468  alephprc  8501  alephfplem1  8506  alephfplem4  8509  iunfictbso  8516  unctb  8606  r1om  8645  cfom  8665  itunifval  8817  hsmexlem5  8831  axcc2lem  8837  acncc  8841  axcc4dom  8842  domtriomlem  8843  axdclem2  8921  infinf  8962  unirnfdomd  8963  alephval2  8968  dominfac  8969  iunctb  8970  pwfseqlem4  9061  pwfseqlem5  9062  pwxpndom2  9064  pwcdandom  9066  gchac  9080  wunex2  9137  tskinf  9168  niex  9280  nnexALT  10563  ltweuz  12072  uzenom  12075  nnenom  12090  axdc4uzlem  12092  seqex  12109  rexpen  13961  cctop  19507  2ndcctbss  19956  2ndcdisj  19957  2ndcdisj2  19958  tx1stc  20151  tx2ndc  20152  met2ndci  21025  xpct  27533  snct  27534  fnct  27536  trpredex  29320  bnj852  33979  bnj865  33981 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592  ax-inf2 8079 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-om 6701
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