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Theorem inf3 7338
 Description: Our Axiom of Infinity ax-inf 7341 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7326, 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 7343 and zfinf2 7345.) The main proof is provided by inf3lema 7327 through inf3lem7 7337, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7337. 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 non-empty 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 7328.) 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 7336.) 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 7337.) 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 2285 . . . 4
3 eqid 2285 . . . 4
4 vex 2793 . . . 4
52, 3, 4, 4inf3lem7 7337 . . 3
65exlimiv 1668 . 2
71, 6ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wa 358  wex 1530   wcel 1686   wne 2448  crab 2549  cvv 2790   cin 3153   wss 3154  c0 3457  cuni 3829   cmpt 4079  com 4658   cres 4693  crdg 6424 This theorem is referenced by:  axinf2  7343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308 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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425
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