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Theorem inf3 7550
 Description: Our Axiom of Infinity ax-inf 7553 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7538, 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 7555 and zfinf2 7557.) The main proof is provided by inf3lema 7539 through inf3lem7 7549, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7549. 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 7540.) 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 7548.) 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 7549.) 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 2408 . . . 4
3 eqid 2408 . . . 4
4 vex 2923 . . . 4
52, 3, 4, 4inf3lem7 7549 . . 3
65exlimiv 1641 . 2
71, 6ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wa 359  wex 1547   wcel 1721   wne 2571  crab 2674  cvv 2920   cin 3283   wss 3284  c0 3592  cuni 3979   cmpt 4230  com 4808   cres 4843  crdg 6630 This theorem is referenced by:  axinf2  7555 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-reg 7520 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631
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