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Theorem inf3 8073
 Description: Our Axiom of Infinity ax-inf 8076 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8061, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 8078 and zfinf2 8080.) The main proof is provided by inf3lema 8062 through inf3lem7 8072, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8072. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.  (As posted to sci.logic on 30-Oct-1996, with annotations added.) Theorem: The statement "There exists a nonempty set that is a subset of its union" implies the Axiom of Infinity. Proof: Let X be a nonempty set which is a subset of its union; the latter property is equivalent to saying that for any y in X, there exists a z in X such that y is in z. Define by finite recursion a function F:omega-->(power X) such that F_0 = 0 (See inf3lemb 8063.) F_n+1 = {y y^(X-F_n) = 0, we have F_n+1 = {y m. Basis: F_m proper_subset F_m+1 by Lemma 4. Induction: Assume F_m proper_subset F_n. Then since F_n proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper subset. By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1. (See inf3lem6 8071.) Thus, the inverse of F is a function with range omega and domain a subset of power X, so omega exists by Replacement. (See inf3lem7 8072.) Q.E.D.  (Contributed by NM, 29-Oct-1996.)
Hypothesis
Ref Expression
inf3.1
Assertion
Ref Expression
inf3

Proof of Theorem inf3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inf3.1 . 2
2 eqid 2457 . . . 4
3 eqid 2457 . . . 4
4 vex 3112 . . . 4
52, 3, 4, 4inf3lem7 8072 . . 3
65exlimiv 1722 . 2
71, 6ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  E.wex 1612  e.wcel 1818  =/=wne 2652  {crab 2811   cvv 3109  i^icin 3474  C_wss 3475   c0 3784  U.cuni 4249  e.cmpt 4510  |cres 5006   com 6700  rec`crdg 7094 This theorem is referenced by:  axinf2  8078 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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-reg 8039 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-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  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-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  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-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095
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